The insurance adjustor wants me to sign off on my medical claims. Should I sign the papers and accept their offer?

NO.

Most insurance companies know that you are a little dazed and confused right after your traffic accident. They would like to take advantage of this and have you sign your rights away. Don't do it!  Don't sign anything until your vehicle is repaired and all of your injuries are healed and you are feeling 100%.

The property damage claim is completely separate from your injury claim.  Negotiate them separately.