We have invested more than half our stack on the turn, I don't think we can find here against an unknown.

Is this a HU hand? I guess I wouldn't be folding regardless. I can't say for sure without knowing your 3bet range but I'd guess that AA is one of the strongest hands you show up with here so folding would be a bad idea. Against an unknown it's hard to say what they have. There are straights and sets possible as well as 2 pair. There's also a flush draw, several straight draws plus some Kx that you're ahead of. All these hands could possibly raise the turn. Sometimes people show up with random junk too

The opponent has 88. I posted this hand because my opponents have shown a ton of sets in these situations, and I have been wondering if it is just bad variance.

I don't think you can get away from this spot with such a small stack to pot ratio. Most opponents should have enough worse hands in this spots for you to stack off profitably.