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.

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

. - 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.  

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 

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

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. 

As far as the SD, disperse it as usual...apply any damages and refund the remainder.