Major dry rot/pest issues or perhaps knob and tube wiring would make me hesitant as well. 

I called a company that an insurance had referred to do the drying .  

I’d rather trust a professional with a moisture meter to see if it’s dry (just after or even during a heavy rain would make me feel better) before fixing the ceiling.

Additionally, you should probably instruct the PM to dry out the attic if you have with fans and / or a dehumidifier. 

I’m hesitant because of mold, dry rot, etc. what should my frame of thought be on moving on these.

When a tenant moves out, transfer the deposit back to Checking so it's ready to apply towards expenses or refund the Tenant.If you end up with excess funds in the Checking account, I recommend you transfer it to a third account designated explicitly for future investments.

Just make sure you set enough aside for capex, vacancies, unexpected insurance increase, taxes, tax professionals, and local considerations (excessive heat? 

So today ozone treatment and air return box repair, then I found some dry but soft base board behind the refrigerator

We put a lot of efford and attention into keeping basements dry, but we still have an occasional pipe burst or sump pump fail and see issues and when you get water in a basement; carpet can be dried and cleaned, it's the open cell pad that is holding on to gallons of water like a kitchen sponge.

Is there a world where it makes sense to leave parts of the dry wall up and try to just clean them?