Seems like I'm spamming this place with hand histories lately. I guess when your game hits rock bottom you start questioning everything

How much fear of the flush is healthy in a raised pot, heads up?

I get KK pre-flop, raise it a standard amount, one caller. I'm out of position.

Flop = J 9 rag, 2 spades.

I fire out a pot sized bet, he calls.

Turn = another rag spade.

I check, he checks.

River = blank

I check, he checks, and my hand is good.

Questions:

Considering that the odds of him having a made flush on turn is small, is it a leak to slow down on the turn? Should I be giving him a free chance to get the four to the board flush? Should I be giving him the chance to bet the scare card?

What if I had position there instead? Should I have bet the turn if he checked? If not, should I have value bet the river if he checked the river?