No you don't have to give your insurance info. You can just pay the ten grand yourself if you want.
Your best bet right now is to contact your insurance company, though. Technically they can tell you you're on your own, but hopefully they won't. When they ask you why it took so long to report this accident, and they will, be honest and tell them what you told us. They'll take a look at the other guy's detailed repair estimate, and (because this stuff is what they do for a living), they'll know if he's trying to inflate his damages.
If you try to handle this on your own, you'll be screwed. You don't know how to pick apart a repair estimate, you can't inspect the other guy's damage to see if it's legit, and you sure as hell don't know how to defend yourself in court. Insurance companies know all that stuff and more, which is why they charge so blasted much.
Settling accidents outside of insurance is legal, but it's risky. You just found that out the hard way.