@Michael Johnson
Congrats on the closing, I wish it was under better circumstances. Definitely call your agent and attorney. Worst case you'll have to open a lawsuit against the seller to try and recoup some costs yet the deck is stacked against you.
Cash for keys will definitely be the best route. You aren't just losing monthly rent, you're also putting at risk $10-20k in damages the tenant could do. DO NOT GIVE THEM ANYTHING UNTIL THEY ARE COMPLETELY MOVED OUT.
Unfortunately, squatters are treated as non-paying tenants and we don't live in Indiana so the process is drawn out here. Thankfully the weather is getting better and the moratorium should be coming to a close.
Consult with your attorney, yet if he's willing to pay the rent you're requesting, you might be better off taking the rent and letting him stay, at least until the moratorium is up. I see 2 scenarios here, if we assume this will end in an eviction regardless.
If he doesn't pay, he's a squatter who's treated as a non-paying tenant. If he does pay, then you collect until you decide to not renew his month-to-month lease. You may have to evict him in the end yet at least that's on your terms, and you recoup some costs during the moratorium (as in lessen the amount of payments you miss because of the eviction process).
Remember, the moratorium will end, yet who knows how long it'll take to make their way through all the evictions. Might be a year, no one knows.
Lastly, keep in mind that this is temporary. It's a great learning experience and over the next 5 years, 10 years, etc. the financial burden this will present is simply a blimp. You still made a great investment. Good luck!