I don't think it was after move 27 you messed up. Both sides missed some things earlier, but that's to be expected in a back and forth chess game (most notably 9...d5?? messing up because white wins material by trading everything on d5 since the e-file pin of their rook to your king).
Move 27. Qd2 (the situation you are asking about) didn't have you really do anything "wrong." It's just that you naturally try to attack b2 (the base of the pawn chain), but this is so heavily defended that you aren't really going to easily make progress attacking this target. In other words, you didn't mess up; you just had the wrong plan in that position.
Continuing to aim at b2 is still fine if you use this to get their pieces passively defending and then you switch targets, but the goal isn't to win b2; there's just too many defenders. Other plans could be relocating ideas like ...Rd7 or ...Re7 and trying to advance your central pawns with support of your heavy pieces from behind, or the idea of ...h5 and trying to threaten possibly breaking open the enemy king position.
You were actually doing really well even after move 27. It wasn't move 27 you lost the game; it was really move 34 when you forgot your rook on b3 was hanging. Playing ...Rxb2 is most simple, but ...Rxb4 also escapes as they can't capture without losing their queen due to their pawn being pinned. Of course, this is objective analysis; it's obvious that you probably wouldn't have missed that if you have more than just 4 seconds left and behind on time in the time scramble. 3/2 blitz does have that 2 second increment, but still 4 seconds left isn't a lot with time pressure.
You clearly used up a lot of time trying to force pressure on b2 when it was so heavily defended. I like to pick targets which I see they are unlikely to defend long-term; not just what defends it now, but also what could potentially defend it later. I'll ignore that target if I'm unable to win it, or tie down their pieces to passively defending it.
My recent video just so happens to be about infiltration ![]()
Granted, my game situation was easier to break through than your game was, but you can probably still benefit from hearing my thought-process and concepts like how I try to maintain flexibility. That game of mine also had a similar ...h5 alternative to open up their king similar to what I mentioned in your game; both your game and mine had us choose other infiltration plans though:
In this position, I'm putting pressure on the weakness with three pieces—two rooks and a queen—but my opponent is defending everything. I couldn’t find a plan. What should I have done?
? I messed up the game after move 27.