I've had exactly the same experiences.
I think the key thing is that I'm not that experienced at deep stack poker. I've been mixing some cash sessions into my play to try and improve on that area.
If you're the big stack but the average stack size is around 20BBs, then I would suggest trying the following strategies, though this is by no means gospel.
i) keep playing tight. I'd spew half my stack getting clever with hands like 53s from MP. I've noticed guys with biggish stacks going after the big stack sometimes because of the implied odds
ii) If you're on the bubble / ITM, I try to identify the mid-stack players looking to ladder up and those who're in-it-to-win-it. The former you can shove into a ton and just takes their blinds, limps and raises.
E.g. Let's say villain is in CO w 20BB. Antes are 0.1BB. You're the BTN w 50+BB. You're in mincash territory and he's reasonably placed. MP raises 3x. Presume his range is quite tight: 55+, AJ+, KQs (8.4%). If you shove here with ATC, what will he call with? Assuming your read is correct and he doesn't want to get knocked out, he'll only call with QQ+, maybe AK (2.6%). That's only 31% of the time. So 69% of the time you scoop a nice pot. 31% of the time you're behind, but you still have 25.4% equity in the pot! So your EV is 0.69*(3+2.4) + 0.31*(0.254*22.4 - 0.746*20) = 0.86BB.
Not a bad return for any two cards
This play is still EV for villain up to 25 BB under these assumptions. You can play with the ranges in Pstove, and maybe slice off the bottom of your range, but this is still a very +ev play in many situations.
I've ignored the blinds here; if you have a calling station / super shorty still to act, then proceed with caution, but for simplicity, let's assume they fold to a nitraise + bigstack shove.
If an aggressive shortstack player is stealing your blinds a ton, it's worth standing up to him. He <i>should</i> be tightening up with a big stack behind him, but if he isn't, I'm calling/shoving with a much wider range. Firstly, I can knock him out; secondly, it sends a message that my blinds aren't charity.