The areas probed still seemed solid, but be aware that some types of wood destroying fungal damage will only be detectable once the wood has properly dried out.

I email or text them once again to let them know in writing when I will be filing a forcible entry and detainer (eviction) action and let them know that time is short but they can still avoid the consequences of a damaging eviction filing.

. - Evidence of water damage in the basement. 

A tenant can do $20k of damage pretty quickly and skip town never to be found again, if you aren't careful.  

I’m worried they won’t leave or damage the property.

They did not make an offer to pay rent/damages/legal fees before the deadline of the eviction. 

But that means we have to push this out even further and the longer the issue isn't addressed, the more damage that is being done.

I would get a professional opinion from a qualified contractor , you may have storm damage already that could possibly be covered by your insurance 

If you buy the next deal correctly you won't damage the inherited property or kill the current cash-flow.

So basically I'm at risk of the tenant damaging my property and not having a way of going after her because she is not signed to the lease.