Overpair on drawy boards

This isn't my hand, but it's a hand that I read on 2+2

http://forumserver.twoplustwo.com/39/small-stakes-pl-nl/100nl-aa-kq9-fd-vs-check-raise-406171/

Basically people are saying that it's a 'coinflip or crushed' situation. And I can definitely understand this point of view, that if you get it in on this flop - villian is always going to have a decent amount of equity.

However, I have almost always wanted to get it in on these kind of boards, simply because it's so drawy and therefore I'm in better shape against his range.

If I should be folding flops like the one posted, then when on earth can I get it in? I previously was more inclined to fold on drier boards, because he's almost always 2pair+ and there are no draws for me to include in his range that I am ahead of?

Id be stunned if hes folding after that raise size, he's got a pretty polarised range between decent draws and nuts here id imagine. That said, even against the draws, you really only have ~50% equity.

The possibility of a straight on this board, makes a massive difference to this hand, and makes it a hell of a lot closer with AA than on KQ8. The addition of J10, which is 16 combos and against which you have ~0% equity is basically the difference in this hand. If we compare it to KQ8, its a pretty big difference, not only is there way fewer nut hands, but way more draws:

These ranges are prob not completely accurate if his 3betting stat is close to accurate, but it doesnt make too much difference to the basic idea.

Board: Kd Qc 8d
Dead:

equity win tie pots won pots tied
Hand 0: 65.032% 64.96% 00.07% 17365 18.00 { 88, AdQd, AdJd, AdTd, Ad9d, KQs, K8s, QdJd, QdTd, Qd9d, Q8s, JdTd, Jd9d, Td9d, KQo }
Hand 1: 34.968% 34.90% 00.07% 9329 18.00 { AcAh }


Board: Kd Qc 9d
Dead:

equity win tie pots won pots tied
Hand 0: 78.162% 78.11% 00.05% 27065 18.00 { 99, AdQd, AdJd, AdTd, Ad8d, KQs, K9s, QdJd, QdTd, Qd9d, Qd8d, JdTd, KQo, JTo }
Hand 1: 21.838% 21.79% 00.05% 7549 18.00 { AcAh }

The way i'd solve it would prob be to assume that he's never folding [which he isnt after that raise size]. If he does, great, but it doesnt make a huge difference. So we need 46.2% equity to continue and even against the most optimistic range [where he 3bets all his nut hands]:

Board: Kd Qc 9d
Dead:

equity win tie pots won pots tied
Hand 0: 59.047% 58.96% 00.09% 8755 13.50 { 99, AdTd, Ad8d, Ad7d, K9s, QdJd, QdTd, Qd9d, Qd8d, 8d7d, 7d6d, 6d5d }
Hand 1: 40.953% 40.86% 00.09% 6068 13.50 { AcAh }

We dont have that. We really only have 46% equity against only his draws, where he has 0 value bets and even then its close. I think id normally be calling the flop and making a decision on the turn. But looking at pokerstove, im not even sure thats correct. The difference in equity from the flop to the turn actually decreases on a blank [wtf, i cant understand that]:

Board: Kd Qc 9d
Dead:

equity win tie pots won pots tied
Hand 0: 77.936% 77.91% 00.02% 30082 9.00 { 99, Ad8d, Ad7d, KQs, K9s, QdJd, QdTd, Qd9d, Qd8d, JTs, 8d7d, 7d6d, 6d5d, KQo, JTo }
Hand 1: 22.064% 22.04% 00.02% 8510 9.00 { AcAh }


Board: Kd Qc 9d 2h
Dead:

equity win tie pots won pots tied
Hand 0: 78.963% 78.96% 00.00% 1355 0.00 { 99, Ad8d, Ad7d, KQs, K9s, QdJd, QdTd, Qd9d, Qd8d, JTs, 8d7d, 7d6d, 6d5d, KQo, JTo }
Hand 1: 21.037% 21.04% 00.00% 361 0.00 { AcAh }


Before i did all this^ i was thinking i wanted to stack off by call/call, but it really doesnt look like that anymore. If he made a smaller c/r, id prob call and fold the turn, because i think he'd have tons more pair+BDFD, or gutshot kind of hands which we can bluff catch against. But i dont think getting it in here can be good. Its just because he has so many nut hands because of J10 really.

Thanks for the lengthy explaination I appreciate it. So basically because of his bet sizing we polarise his range to a very very strong one.

So let me give another example, say one where we have AA on a 8d6c4d board.

Are we now stacking off because of

1) less likely to have a straight (altho it is bond)
2) Can have other overpairs, which we crush
as well as obv including sets/2pair and fds in their range

And am I right to be wary of stacking of on really dry boards still with overpairs? Because of the lack of draws, villains range is 2p+ or bluff?

Also, I would imagine that the equity goes down on the turn blank because we have actualy have more combinations of hands that beat us because there is only one combo of A8dd and QTdd etc?