Buyer wants price adjustment after closing paperwork has been signed?

tl;dr – selling home, did everything asked, signed closing paperwork, now buyer wants more money off deal

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I'm selling my house, and someone put in an offer about 6 weeks ago. It was listed at a very good rate because the HVAC is old – it still functions perfectly fine and had routine maintenance, but it's 20ish years old and we were up front about it's age the whole time. I've had everything fixed that was on the home inspector's list with no hesitation, and I even accepted a small negotiation down on price due to the HVAC age plus paid to have the unit serviced.

We went through everything we were asked to do and were on pace to close a few days ago, then they pushed it back…after they'd already sent and had me sign the closing paperwork (I've moved states), and had it notarized and sent back to the buyer's lawyer.

Now the buyer has come back again wanting even more money off for the HVAC (to a point where all the concessions nearly match the price of a full new unit installed). Does the buyer have any right to do this, and is the closing paperwork not legally binding? Never sold a house before but it seems like an offer was made, everything that was asked was done, and their lawyer sent me final paperwork to complete the transaction, so it should be done.

I'm aggravated and it seems both my realtor and the buyer's realtor aren't happy about it either. I would like to have it done with so I can stop having to scrounge up to pay for two residences while getting this sold, but I don't want to get taken to the cleaner either. What are my options?