A Labral tear in the glenoid (soft cupping tissue for the shallow ball and socket joint which keeps the humerus from dislocating) can be improved by immobilizing the joint in a sling. If you've already experienced dislocation of the shoulder...called a Bankart lesion...it might be best to get the surgery. Chances are another dislocation will occur at a very inopportune time and place.
Otherwise...you could immobilize the joint in a sling, take NSAiDs meds, and hope for the best. But if surgery is already decided...it's probably too late for immobilization to help because the Labral cuff is compromised pretty bad.
The recovery period is long w/surgery...6 mos. if I recall correctly...maybe shorter w/arthroscopic surgery. Maybe a second opinion would help you decide ??
Personally, I would have the surgery to prevent further damage by repeated dislocation and possible arthritis, and look to the next season. And...what if the shoulder dislocates while at full draw ? Ouch !...and it could be disastrous.
Best of luck in whatever you choose, though.
Mission
