I will start with
If it ain't broke, don't fix it
In your body question, you state
The players seem to be having fun, but the rogue's already been rendered unconscious once.
If the players are having fun, and assuming you are having fun as well, things seem to be going fine.
They are at level 3 and the only thing that worried you until now is that the Rogue has been rendered unconscious once? My current group of players has 6 players and in their first fight an incredible count of four characters dropped to 0 HP and two of them only didn't die because I was being kind. And that was supposed to be an easy encounter.
The point I am trying to make here is you seem to be overthinking. The players are having fun and getting unconscious is 3 failed death saves away from actual death - just don't attack them while unconscious, if that's what worries you. (I linked a question about attacking unconscious characters.)
DMG has guidelines on it and you are following them
(You mention it in a comment that you are aware of the DMG guidelines, so I'm not stating them again)
From my experience, these guidelines are actually "easier" than they should be. A balanced party can take alot more than what the DMG states, if their encounters are regular (i.e., it's not full of surprises for enemies, enemies don't have terrain advantage, etc).
Again, don't worry too much, from what you've said until now, you're doing fine.
As Premier Bromanov already answered, Healing is not exactly a must in D&D 5e. If you are still worried about it, check Optional rules for a DM to compensate for lack of healing in party? - one additional thing that you can do, mentioned in the body of the question and usually my own approach in games where I notice lack of healing from the party, give them a little more potions of healing than usual.
Your question seems to be worried about the number of encounters from two perspectives: one is the mechanical one, where you mention that you have a poorly balanced party and the Rogue got to 0 HP. I have answered this part so far. The second part is about the feeling from the players, where you mention
I don't want the game to be all about just smashing monsters… or, at least, I don't want the players to think the game's all about smashing monsters!
For this part, Premier's answer has done a better job than I can think, so I'll just repeat my initial statement: If it ain't broke, don't fix it. Your players are having fun playing what seems to be a game of smashing monsters - many people love dungeon crawling and D&D was actually born from it.
If you actually want to change it, then proceed with Premier's bullet points for creating a more role-play oriented adventure or session. We have some other questions that might help you with that as well.
- How to get the players to care and RP more?
- How do I get my PCs to not be a bunch of murderous cretins? (I think this is one of the most popular questions in the site, if no the most.) - This one might help you include some guilty feeling even in smashing monsters! They are living creatures as well, you know? Entering their lairs and merciless killing them is not a nice thing to do.
- In general, reading over the [dnd-5e] + [gm-techniques] tags (Query here) helps you as a new DM. I have been DM'ing for years before knowing RPG.se and it helped me with problems I wasn't even able to identify before, but existed.