Talk to Bobby Hoyt about relining your shotgun barrel. There is also a company out in Sprague River, Oregon, called, " Mark Behnke Gun Shop", that does barrel relining. He specializes in liners for original Springfield cartridge rifles. Bobby is a member here, so check " members " in the bar above, and find his contact information. There are a couple of other barrel people around the country that also do relining.
Whether a gun barrel can be relined depends on the condition and thickness of the barrel. If the barrel in its current state is too thin, you probably won't have any choice but to reduce the bore diameter to a smaller gauge. That will affect the balance of the gun, and you should consider that factor in making your decision to proceed.
I have a personal friend who had his family heirloom L.C. Smith Damascus DB shotgun resurrected by having the original barrels cut off in front of the hinge, and the chambers, and then having the chambers " sleeved" with two new barrel extensions. The sleeves allowed the barrels to be rechambered to the same 12 gauge, and the new steel barrels allowed the use of modern ammo.
A machinist did that work for him, and I suspect a skilled gunsmith could also do that work. The cost will be a factor. My friend was willing to pay the extra money because of the heirloom factor involving the gun. Without that, he could not have justified spending as much on that work as it would cost to buy a new gun!