The other driver's insurance company wants me to sign a medical authorization to get my records and bills related to injuries sustained in the crash. Should I sign it?

Sign it only if you don't care if the insurance company looks at all of your medical records.  The insurance company usually sends a broad  medical authorization that will allow its employees to get any of your past records along with your current treatment records.  This obviously allows them to get records that are private and have no relevance to the injuries sustained in the auto crash.  

By signing the authorization, the insurance company also now has permission to talk directly to your treating physician.  The insurance company may also ask your doctor to fill out a questionnaire in which he or she is asked when you will likely complete treatment.  If your treatment takes longer than what the doctor estimated, the insurance company may claim that you are exaggerating your symptoms.  

The better rule is not to sign the medical authorization until you have consulted with an attorney and discussed all of the ramifications.