[Nougat]

If I'm not mistaken, you're getting 3:1 odds to call into the pot on the turn. Your hand at that point has the 9 spades and 6 aces and kings as outs (assuming they're all clean which in this circumstance is a reasonable read. His play screams PP anywhere from 99-QQ generally depending on how he's been playing. We can discount one of the spade outs to account for the paired board and the possibility of him getting a full house down to 8 spades. So it looks reasonable to determine your outs as anywhere from 13-15 outs on the turn. Even with 13 outs, your call is marginally correct.
I think it's pretty clear you have little FE for a shove on the turn based on how much he has behind after his bet. Shoving on a semibluff is +EV if your FE is approximately 9%. If you have 15 clean outs, then your showdown expectation is ~435. With 13, it is -1348. Therefore, using the most conservative measure of only 13 outs, you need him to fold 9% of the time for this to be successful. So you can shove on the turn and still get value. But I think a call is better (please someone correct me if I'm wrong politely).
If you call a few good things for you can happen:
1)the guy behind you might call, which gives you even better value.
2)the guy behind might raise in an attempt to squeeze your money but you'll still be getting great odds to call him too even in a protected pot.
3)the guy behind you might fold in which case your variance will decrease and you've made the greatest EV decision in a HU pot.
On the river, it should be plain to you that he has an overpair by how it's been played thus far. If it's a bluff (and a maniacal one at that), he'll need you to fold more than 70% of the time in that spot. Obviously that won't be the case and it's a losing play for him to expect that. In this one case you will fold the river, but in the case he's bluffing (truly rare by now) then you will also have to do something improbable (your 30% fold) for him to show a profit. I think calling and then folding a missed river is the greatest +EV and with your stack as deep as it is relative to the blinds I don't worry about being totally handicapped. Comments, thoughts? Lemme know.