A lot of the things that will make you more money at 5nl will cost you money at other stakes.
Personally after a sample of... dunno, 30k or so, I've averaged 28/26 and have started playing 34/30 (both 6max) and feel like I'm killing the game, but my winrate isn't much better than yours. As for breakeven stretches, none, if I could get 100 hands without a 100bb swing in either direction I'd be happy. Had a few major downswings, -4BI a couple of times, -10BI twice (yikes).
Yeah, I do suck that bad at poker.
To be honest, it sounds like you've had a similar run to me (with fewer swings?) and you should be way over rolled for 10nl by now, if that's the case, move up. If you're beating 5nl for a decent winrate you'll most likely beat 10nl too. I don't think you're missing out on developing any fundamental skills at this point, just move up and fix up any leaks as you go. I mean, if you've got a decent pre-flop game then that's about all there is to learn at these stakes since hand reading/metagame etc play a very minimal role and because of that the post flop play is over simplified.
Your winrate also could just be variance. I have no idea on the odds
of it, but it's common wisdom that 3PTBB swings from your true winrate aren't uncommon up to 100k hands.