The tenant has not told about any water leak or moisture issue or mold till date.

Sounds like you are doing the right thing with the moisture management.

It's not going to matter unless you have some kind of weird vapor barrier preventing moisture movement below the sheathing - water vapor moves through the roof sheathing all the time, and wet wood will dry out the same way (water vapor moving from wood to drier air). 

My lender heard from the appraiser that there was moisture in the basement of a couple places I am trying to refi, and all hell broke loose.

I would have concerns about this anyway, but do so especially because there has been moisture in the house and the attic is not vented. 

.,: "rolling" foundation walls; possible exposed asbestos in a small area that could cause it to be vented into the house through the HVAC and possible mold due to all the moisture. 

@Holly PetersonThe hardest part is really keeping the water and moisture out of the basements.

Should we be investing in moisture meters and other things?

It only grows when it has access to moisture and a food source (dust, food, etc.) and will go away if you remove either of those.

On a slightly more serious note a large capacity fan that vents outside tied into the light switch so when the light is on the fan is on is a good way to move moisture out of the room.