A Drilling is a rifle under a double barrel shotgun. You can't convert a rifle barrel to a shotgun barrel! Unless you talking smaller than a .410 ! You could replace a rifle barrel though.
Most of the Drilling rifles were chambered for a 9mm rifle cartridge, or 7, or 8 mm. I forget the correct designations for those calibers. as for stocks contact Pachmayer . I believe you can still get fancy grade french walnut shotgun butt stock blanks from Dixie Arms. I don't know what Pachmayer will charge, but I doubt it will be cheap. But if you want to do honor to a fine piece of old world craftsmanship, then that would be the route to go. When I was in Italy, I was in a small gun shop, and admiring the beautiful shotguns this guy and his Dad made. The Bluing was just phemoninal! He had a beautiful .410 side by side with hammers. I almost bought it! But I was in the army, and at $105.00 a month, I couldn't afford much! But I got to talking to the guy, and he was showing me some of the guns he was building in the back of the shop, and some he was repairing, and he showed me a beautiful side by side 12 gauge, and asked if I knew of Pachmayer, and I said yes, he said well he made guns for him, and showed me an order form from Pachmayer, and in fact this was one that Pachmayer, had stocked! Check them out and see if they will make a stock for it. If it was made in the 20's, it should handle smokeless powder. Are the chambers able to accept a modern 16 gauge shell?
Or are they the shorter shell? Some of the late 1800, and early 1900's on 12 gauges at least had a 2 1/2 inch chamber. But check with Pachmayer, and see what they say, or suggest. Also check out Imperial, and Bishop. Or just Google gun stock makers, there a bunch! Good luck. Beautiful find!
You sound like Eddie and his flea market finds!
Wayne