If the 3rd party insurer has admitted liability (which they'd be mad not to) they will probably want to settle it directly with you, avoiding your insurance company. Bear in mind that this route will save them lots of costs because if a claim is made on your insurance, then repaid by the 3rd party insurer, your insurer will add their costs to the final bill the 3rd party pay.
If they make you an offer directly, therefore, you are in a strong position, because, if you say "no, thanks, I'll deal with my insurer" their bill immediately grows considerably.
I would take it to a repairer of my choice, get them to quote for repair by the 3rd party insurer and say take it or leave it. If they start trying to claim it's a write off, reject any settlements they offer unless they are very generous, and start making noises about getting your insurer involved, getting your insurer to provide a courtesy car, etc. and they will start to see the costs spiral.
A 3rd party is indeed liable to put you back in the pre-accident position and can't declare your car a write-off unless you agree to it. That said, they won't pay a fortune to fix an old car, but, under threat of involving your insurer you should be able to get a pretty generous settlement directly out of them.
Don't let the 3rd party insurer take your car or any documents unless you've agreed to let them write it off.