Truly donking the nut flush

[Jiminy]

PokerStars Game #3329609950: Hold'em No Limit ($10/$20) - Seat #5 is the button
Seat 3: Hero ($4353 in chips)
Seat 4: xxxx ($1941 in chips)
Seat 5: xxxx ($1281.40 in chips)
Seat 6: Villain ($5830.50 in chips)
Seat 8: xxxx ($4065.50 in chips)
Seat 9: xxxx ($3874.90 in chips)
Villain: posts small blind $10
xxxx: is sitting out
xxxx: posts big blind $20
xxxx: sits out
*** HOLE CARDS ***
Dealt to Hero [Ah Kh]
xxxx: folds
hero: raises $30 to $50
xxxx: folds
xxxx: folds
villain: raises $150 to $200
xxxx: folds

hero: calls $150
*** FLOP *** [9s 9h 2h]
villain: bets $400
hero: calls $400
*** TURN *** [9s 9h 2h] [3h]
villain: checks
hero: checks
*** RIVER *** [9s 9h 2h 3h] [5h]
villain: bets $400
hero: calls $400


Please, berate the way i played this hand from start to finish. I have been sitting at the table for a few hours already and have not seen villain do anything out of line. He has not been overly Aggro or anything like that. I realize that in order to call a pot sized bet on the flop, I have to think i have big implied odds if i hit my hand, but then when i did hit it i froze up, afraid of the pair on board. When the turn came a heart, I felt that villain would read a flush pretty easily so i decided to check behind and make it appear like i had a medium pair that i wasnt ready to let go of on the flop, something like 88. I was then going to get aggressive on any non broadway river <my read was easily that they had an overpair, so if the river coulda filled them up i woulda played it safer> The river bringing a 4th heart changes my plan. Since i have both the A and K of hearts, the best hand they could have that isnt a boat would be QQ with Q hearts, and i doubt that they would call a reraise with that. Since i thought they wouldnt call the reraise, i didnt want to open the betting back up to them incase they did have an awkwardly played 22 33 or 99. How should i change each street here. How can i play a draw like this on a paired board when stacks are 200bbs deep and i have a lot to risk if they have a boat. Is this an easy call, an easy muck or an easy raise on the flop?