On top of that we will get a judgement for money owed.

Was this poor judgement on my part?

If they don't they will have to pay that rent or you can seek a judgement against them. 

A person can literally own a million dollar home free and clear and you cannot touch it-Texas does not allow garnishment of wages for a judgement so that avenue of collection is out-You can get a judgement and hope or even use a collection agency who will take 50% or so if they are successfulOR you can take the money now and move on and get a new tenant. 

I make mention in my ads that if they have unpaid landlord judgements that they’re wasting their time even contacting me.

My understanding is that even though there is a moratorium, the tenants are still liable for the debt. Has anyone successfully been able to sue a tenant for back rent? I realize that collecting is another matter, but ...

Questions like this are exactly why you should have an experienced agent representing you and watching out for your interests.A title search performed by a title company or real estate law firm determines the vested owner, the liens or other judgements on the property, the loans on the property and the property taxes due.

Not nearly enough info here to make a judgement call...talk to them and find out what the deal is...post it here and you might get better feedback. 

For example, if you have a judgement against you of $500,000 and you only have $300,000 coverage on the policy for that property, the umbrella would come in after $300,000.  

Now all moneys must be paid up to date or I will be forced to file for a judgement which will ruin your credit .