I'd be folding flop 100bb deep, seems borderline to call even with our backdoor draw and overcards, think this deep it's fine though. Turn I call again as our draw is hidden and I expect we can take their whole stack if we hit our flush.
Again 100bb deep I might fold the turn, 150bb, calling flop and turn.
Flop: I like check calling here. We're going to have the best hand here sometimes and it's going to be very difficult to fold out better hands here so I don't like C-betting or check folding.
Turn: With so much equity I would just be check calling again. We can't rep a huge amount by check raising here and I would expect to get looked up here quite light fairly often if we opt for this line. I think with our overs + flush draw + potential chop outs to the straight we have enough equity to continue though.
Check calling this flop texture would be my standard, I expect to have the best hand a ton when we squeeze a CO open from a 27/21.
I think we can check call even a decent size bet on this board with so many outs. for about 1/3 pot it looks straight forward enough. We can donk an A or K river and expect him to put us on a bluff as it's only this exact combo that picked up the flush draw that ever check calls twice and hits those cards.
I agree with Ferg that check raising the turn doesn't really work on this board because it's tough to rep anything.
No decent reason to not cbet here. Check/calling is reasonable at least, but usually not the best play.
As played, check/jam turn. Should be pretty straightforward really. It really doesn't matter if we get looked up light when we have two overs and the nut-flushdraw. We might even get called by worse.....