What recourse do I have against the oil change shop for damages if it overfilled the transfer case causing the entire system to leak?
You can sue the oil change shop for negligence. Negligence is the failure to exercise due care (that degree of care that a reasonable oil change shop would have exercised under the same or similar circumstances to prevent foreseeable harm).
In order to prove negligence, you will need to prove duty (of due care mentioned above), breach of duty (failure to exercise due care), actual cause, proximate cause and damages.
Actual cause means but for the oil change shop would the oil have leaked? If the answer is no, which appears to be the case, actual cause has been established. Proximate cause means were there any unforeseeable, intervening acts which would relieve the oil change shop of liability? If the answer is no, proximate cause has been established. Damages means the amount of compensation you are seeking in your lawsuit. Your damages would be the cost of repairs. You can file your lawsuit in Small Claims Court. Your damages should also include court costs which would include the court filing fee and process server fee.