I think Im raising the flop. Most players would bet bigger with a good hand here, you have no SD value and there are tons of worse hands he will just fold to a raise. The K and the T will be great cards for barreling putting him in an ugly spot even with his 2pairs. As played Im raising the turn as well. Its kind hard to rep a bluff here though, you are rarely calling that flop with air and would now have to be turning a pair into a bluff which he shouldnt expect too often.
But I think there are simply enough hands that you can get value from now, such as any set and any straight, good 2pairs might make a bad call as well. Youre hand is vulnerable as well, any spade might kill your equity or your action (+ the 3 non spade kings).

Fold pre, you're leaking money

Post flop is well played. I wouldn't raise OTF because I don't think you have enough fold equity on this board, since villain probably connected. Pre-flop I would either 3b or fold, don't like calling so much, even in position, because you are gonna have 0 equity in tons of boards. I would prefer flatting J8s (and that type of hand) than 56s.

I'm also folding preflop. As played you played it fine.

I'm 3betting that hand, but as played I'd rather raise OTT because if another spade comes I would punch my monitor.