I am currently running a D&D 4E game. The game started off kind of goofy (we have a thrikreen rogue with glowing blonde hair, an ale-obsessed dwarf battlemind, a ditzy human wizard, and a dwarf shaman who doesn't like nature), however, I want to tell a more serious story without having to refer to Sham-wow the Shaman. What is the best way to get the campaign into a more serious mood or atmosphere without ruining the goofy fun my players are having?
Talk to your players
It's entirely possible to have Serious Events in a story that's still a bit silly (works like Discworld and Digger attest to this), but it requires tonal shift that everyone has to buy into. GMs are not the Fun Police, and if your group doesn't want to play in a more serious game there's not much you can do to force the issue.
If you can convey to them how excited about it you are, and ask for their help, then this will be something everyone is doing together: not you spoiling their fun with Serious Business.
If you've got player buy-in, then consider one of these options:
Take a break from the light-hearted characters and run something grittier for a little while
Whether in the same or a different system, a serious interlude with new characters might be more welcome than asking them to abandon their beloved PCs entirely or adapt the silly PCs to an unsuited shift in tone. If everybody has a good time, some serious elements might be more welcome in the main campaign afterward.
Run an adventure around risks to things the PCs care about
This can go terribly wrong if the players don't buy into it, but if they do it can be a great way to get more player investment in the world. You can work with the players to identify things, people, places, or values that their characters take seriously, and then you can craft an adventure where those are what are at risk (rather than PC death or not getting more loot being the main risk). This gives the players a chance to explore their PCs in more depth without derailing the funny parts of the campaign: even funny characters can be well-rounded and have things they take seriously.
By and large, you don't. I enjoy goofy, the rest of my group prefers dark and gritty. We end up playing a campaign of dark and gritty, followed by a few sessions here or there of goofy.
From my experience, a serious character can become goofy much easier than a goofy character can become serious. Like you mention, if the shaman is named "Sham-wow", people will chuckle/snicker every time they refer to him/her. However, Grognard the Brave can just dye his hair dayglow green and cut up.
To better continue forwards, your current game should be allowed to reach a natural pause/end point. Then have the players roll up new, "serious" characters. However, there will never be an end to goofiness. People mishear things and will laugh. For example, an NPC in my most recent session I swear to you was named "Viagra". Cue the many double-entendres based on THAT name. We had a few jokes at this poor NPC's expense, then got back into character and continued with the serious storyline.
In general, find new players and start a new campaign.
If your players are the types for silly names, you're not going to get them to do serious for very long.
I want to tell a more serious story without having to refer to Sham-wow the Shaman.
This tells me that, unless the game was intentionally humorous, your players are the silly types to begin with, and will derail any "serious" game sooner or later. I've known several such players, and they seldom deviate.
Some will deviate for a while for a new setting, but sooner or later, well, the knight with the Pepsi Symbol for his arms...
Take a note from real life, people may be dying of some tragic disease and yet still encounter funny things, it works both ways.
One of the best DM's I've had did exactly that and this is how: Our good characters started out as kids excited to go on an adventure (our "Sauron" got nicknamed John Holmes) we always got stuck with ponies (which apparently we could wear as shoes or pick up and carry under our armpit if wading across a river) and then one day we had to pass through a mountain and a wizened old guardian and as a test of our goodness we had to give up what we cared for most (one of us lost their minion, one lost their awesome sword and the other lost all their hoard of moneys) and we and our characters had to grow up a little... then we got back to taverns and drinking and doing improper things until one of our characters ended up having to do some evil errands eventually 2 characters became evil (and in the epilogue of our campaign "their names were used by mothers in centuries to come to scare little children into going to bed") and even when they were evil they managed to stuff a goblin down a dwarf's plate mail and he suffocated from the smell... The rest of the group managed to save face and become honorable heroes who were not remembered as long as the evil ones but who did managed to save the world from the eeeeeevil jooooohn hoooolmes! Life may be silly and yet serious stuff still happens... the opposite is true as well ;) and just have fun.. seriously!