I empathise! You're not alone or weird for feeling like this: I'm also the kind of GM and player who enjoys deeply engaging with the game, and I find freewheeling silliness to interfere with my enjoyment of the game. Few people understand why they're incompatible for me, but I've learned that I'm not alone and I can share that knowledge.
The good news, then, is that the problem isn't you. The bad news is that the problem isn't them, either. The problem is that roleplaying is an incredibly flexible pass-time with a multitude of possible "payoffs" for spending time on it, and what you all want out of a roleplaying is incompatible. There are many ways to enjoy roleplaying, all of them perfectly normal, but not all of them compatible. Trying to combine people who enjoy silliness in their roleplaying with people who enjoy serious, silliness-free play just doesn't work.
(Aside, I'm going to use "serious" throughout this with some caveats. "Serious roleplaying" is usually said when the speaker means taking engagement with character, or plot, or embodiment of character seriously. It's easily misunderstood though, since people who don't "play serious" can very easily be taking other parts of the game – tactics, character advancement, fun – just as seriously. When I say "serious" I'll try to put it in scare quotes, because I don't want to devalue or denigrate what others take seriously, just because it's different than what I do. If I miss a scare quote it's an oversight, not a slight, and I'm still meaning what I'm saying in this here parenthetical.)
I suspect that your other problem – not feeling like you can speak up about / enforce your desired tone when GMing – is a side-effect of the mismatch between how you want to play and how you perceive your group wants to play. Few people enjoy conflict, and fewer have the confidence to calmly champion a serious, character-engagement style of play when their friends make them feel like it's the weird way to play. It's doubly (or is that triply now?) harder to do it when you don't get the chance to practice and feel confident about playing that way. On top of it all, you clearly do enjoy the puns – I bet you wouldn't feel bad about your own punning and participation in the silliness if you ever got your own style of play desires satisfied. It's easy to feel like you're indulging in a "weakness", and blaming that for why you don't get to play the way you want, when the real problem is elsewhere and non-obvious.
So, what to do?
I have a three-part prescription for you:
Be a player
This is about getting that confidence in embodying a character. The idea isn't to be perfect, but to just get some hours on your timesheet as practical experience. It's time to make some missteps, make some discoveries, and level up your personal roleplaying skills a bit. It's also to just get a back a feel for roleplaying without the responsibilities of GMing getting in the way. It can sometimes be hard to get into character as GM because you're constantly switching characters, and there are so many other details that need attention and pull you out of character.
Suggest that someone else in the group run a mini-campaign for a change of pace. Find another group in the area that you join as a player. If it suits your temperament (it doesn't mine) do some play-by-post or play-by-chat. The important thing is to lay down your GMing burdens for a while so that you can just roleplay, take some pressure off yourself to be The Person Responsible For Group Enjoyment, and rediscover what having only one character is like.
That's not going to do much to get you your serious gaming fix though, unless you get lucky and land in another group that happens to be all like you. So the next step is…
Find a serious-roleplay group
Why not try to convert your group to serious roleplay? Because making someone play in a way that is contrary to what they enjoy in roleplaying is going to make them unhappy – as you well know. Trying to overtly or covertly convert an existing group to a different playstyle without their full and enthusiastic buy-in is the surest way to destroy a group. The way to have a group that does play in the style you enjoy is to form it or find it.
If there is a subset of your current group that is "more serious", invite them to play in a side game. Say that you want to play a game that emphasises character and roleplay; if they're enthusiastic, run with it, and if they're not let it go. You're testing the waters here, seeing if you have kindred spirits among your existing group who might indulge more in the character-engagement side of things in a different group dynamic. Don't push though – if they want it, they want it. If not, you'll just be doing a lot of work to set up a group and a game only to get no return.
The surest way to have a group play in the style you enjoy is to find one that's already formed, but it's also the harder way. If you can't make one though, the harder way might be the only way. How is beyond the scope of this post, but "Where can I find other RPG players?" might help.
You can combine finding and forming: if you do get some interest from a subset of your group, joining another group together can get you the benefits of more serious play and the chance to be a player.
The point of getting into a group that enjoys more character engagement is to scratch the itch for the roleplaying payoffs that you're just not getting now. If you're not feeling like something is missing all the time, you will probably find that you will better enjoy the things that your existing group's playstyle does have to offer you.
Find a "serious" RPG (optional)
This one is optional, but has the possibility of being the most effective. Everyone's mileage will be different though, so this really is optional.
Not all games are made equal. I don't mean that some games are better designed or written than others (which is true but irrelevant), but that different games target fundamentally different styles of play. I don't know what kind of game your group is playing, but I'm going to bet it's one of the myriad so-called "traditional" games with rules that tell you how to resolve a sword-swing or a long-jump over a chasm, and how to improve your character's abilities, but leaves the what and why unspoken and up to the group to figure out. There are games that don't follow that tried-and-true mold and try to do something different – something you might like. Nobody plays an ongoing campaign of Dogs in the Vineyard for lols.
There are too many to list or even shortlist, but some keywords to start googling and investigating are "indie rpgs" and "small press rpgs". There are tonnes out there and it seems like there's a new one published every week, so you've got the blessing and curse of a lot to choose from. (On the plus side they're usually self-contained in a single book that doesn't cost an arm and a leg, and many are available even cheaper in PDF. Some are even free.)
The point of this (optional) step is to find a game that actively supports your creative agenda: deeply-felt roleplaying with extensive character embodiment, or whatever metagame rewards you've identified are your priority. It's non-obvious how that's even possible when all you've known are traditionally-structure games, but the last decade of cutting-edge roleplaying design has been in the direction of how to build mechanical foundations that better support specific, under-served playstyles such as yours. I don't have much specific advice for what to look for or how to find it, except go forth and discover this frontier of RPG design. It's a journey that is different for everyone.
But what to do when you've found a game or games? Read it, for starters. There are so many that you'll probably find way more that are interesting than you'll ever be able to play, but you can still learn a lot from reading them. Some of them will contain ideas that will change the way you run and play games for the better, even if you never run or play that particular game.
Second, suggest a game to your existing group. Do not under any circumstances force it on them, because buy-in is really important for learning a new game, let alone one that offers a new way of playing. (I did that, oops. I don't have that group anymore. Learn from my mistakes!) If they're curious or enthusiastic, offer to run (or "facilitate", if that game doesn't have a GM) a single session. One session of a new, weird game is much easier to commit to than an indefinitely-long campaign. It also lessens the chance that someone will say "yes" to a proposed campaign just to go along but will end up (un)consciously sabotaging it to get back to the "regular" game. Taking a chance on something new for one evening is something that most players (though not all) are willing to do. If they're on board, great, if not, you haven't lost anything for asking.
Thirdly, you can form a group around one of these games. Some subset of your existing group might be curious, even if the group as a whole isn't. Also, in my experience some of these indie/small-press/alternative games are more attractive to non-roleplayers than to established roleplayers. You might find people in your non-gamer social circles who would never play D&D and automatically dismiss anything if the word "roleplaying" is mentioned, but who would like the idea of one night hanging out in the living room for a game about playing a TV show. A lot of people have preconceptions about roleplaying games that are based on traditional games and traditional groups. The fact that these other games are so different is something that can actually be an advantage when proposing an evening's entertainment to a non-gamer friend.
Fourth, you can find a group that is already interested in these kinds of games. The kind of games you're interested can be a kind of plumage: interest in alternative games tends to say something about what kind of playstyle you enjoy and attract kindred gamers to you. The thing about popular games is that everyone and their grandmother plays them, so you can never really know what kind of game actually happens at their table. With indie/alt games, you can have a much better idea of what people's play priorities are based on what games they like. Find the games you enjoy, and follow those to the people who also enjoy them.
Again, looking into other games is totally optional. If it appeals to you, awesome. If it doesn't, then awesome. I can only say that investigating them has done a lot for my own understanding of what I enjoy, why, and how to get it, but it's such an individual process that I can't prescribe it as the universal solvent for playstyle conflicts. Alternative RPGs aren't a magic bullet.