I ran into a similar spot three times in today's session and I realized that I feel very unsure about this. What do we do with with overpairs in 3-bet pots oop when the board favors the PFC?

My broad take is that in general, we want to check a lot. I understand that checking a decent chunk of our strong hands comes with two advantages for our whole range:

1. When we check, our strong hands profit from the presence of all of our weak overcards in our range that missed. These provide a large incentive of the PFC to stab, drastically increasing the EV of our strong pairs in the checking line. Hence there is quite some EV generated from overpairs by bluffcatching on middle/low boards.
2. When we check, our weak overcards that missed profit from the presence of our overpairs. Since overpairs are in the checking range, the PFC cannot go crazy in stabbing and our overcards realize their equity better.

So do we just check our overpairs and basically range-check on boards like 982ss, T85ss, 754ss? Do we then check/raise or call down? One approach might also be to check overpairs with stable SDV (AA) and barrel through weaker overpairs (KK-JJ).

Here are the three spots. You can see that I'm very unsure since I'm taking a different line each time.