The nursing home that gave care to my dad in 2019 sent me a bill. Advice?

This is exactly why we advise to never sign admission paperwork to a NH stating that you will be responsible for any bill of the person being admitted.

as long as you didn't sign paperwork, the bill is not yours to pay.

Unless you signed off to be personally responsible on your dad’s admission paperwork, it’s not your bill. Any debt he had was his when alive or a debt of his estate. If he died with no assets, there was no estate that warranted opening probate or dealing with MERP (Medicaid estate recovery program) beyond the questionnaire.

so the 8k, why is it really your debt?
and
did you in writing clearly state your responsibility for the 8k to the NH or the debt collection outfit?
id suggest that you ckearlybsieak with your atty as to if any of the 43k or 8k can actually be placed on you first and foremost. Personally I think on your own you can deal with the secondary and third level debt collection bottom feeders IF you respond to them within the 30 day required under Federal consumer laws. If your way way past that, or if it’s gone to getting a judgement against you, yeah, it’s on your atty to deal with it and get it settled,

Yes, working with a collection agency is fruitless. Like Alva said, they buy Accounts Receivables but do not have the details. Its up to the person being collected against to go back to the source and get the details.

I would wonder if he was OKd for Medicaid why the balance owed was not paid retroactively. Medicaid may owe that amount?

Yes. DON"T pay another CENT until you get good solid professional advice from a lawyer. You should not have talked to a bill collector. That you did reopens this to them being able to bother you another 5 years before the debt expires. If you PAY ANYTHING you have assumed the debt. The bill collector bought this paper for 5 cents on the dollar. They are unlikely to proceed against you in court. If they do you can demand fully itemized bills which they almost certainly do not have.
You need to STOP right NOW. Don't make a move until you get the professional advice of a lawyer. You could do yourself thousands and thousands of dollar worth of harm if you do, and for no reason whatsoever.
Take all papers that show if you assumed this debt. Because it was your father getting the care. If his estate has no money it is very unlikely they will enforce any laws about children having to pay. Did you sign something saying you would be responsible for your Father's care.
Seek help. HURRY. You are badly in need a good advice.