OTT bet of give up?

I think you should had to give up this hand turn. Board is very dry and villain call the flop so he have got most of the times TT++, or 77-99.

Maybe some input on your part?
What is his calling range preflop? What hands do you see him call the flop and fold to a 7♥: OTT?

We have just 1 overcard and nut backdoor flush draw. I think we don't have many good barreling cards OTT so we should think if cbetting this OOP is a good idea. We can also visualize villain's range after he did call the big-ish 3bet pre. Pockets and broadways, right? Will he be folding much?

If we go with the cbet, which might not be terrible, then we should give up OTT.

I guess cbet OTF isn’t that bad, this is pretty dry board and cbetting can be fine on those, but if you want to cbet, make it 1/3. You don’t need that much FE to make it profitable, he’ll continue with a bit wider range, so you can barrel on good turns against his wider range and therefore expect more folds. But on this turn, just give up.

But of course checking flop can’t be bad. I think this can be actually a pretty good spot for a check/raise.

A9s is not in my 3bet range preflop but i guess its up to the player what hands to include.

The board is very dry and you can get away with 1/3 cbet here i believe. Villians' calling range should now change if you bet 1/3 instead of 1/2. You can do this with all your hands on these boards including monsters.

I would barrel any diamond turn and an overcard to the T, however, i am check/folding this turn. I doubt he is folding any pair that he calls the flop with.

Preflop is fine. OTF I think 1/3 cbet is much better. That way you are challenging villain´s defending frequencies. On such a dry board, bigger you bet, easier life villain has - imo. In general 1/3 cbet works fine on paired boards, esp on such a dry ones. OTT I would just give up. His range contains now 4x, better Ax, Tx and PP. Basically you are targetting 55-66 + Ax. I think you are separating his range OTF too much and you don´t generate enough FE OTT now.