"Real old"- Serial numbers were not required on rifles and shotguns until 1968, so don't go too much by that.
Yes, excessive headspace may give you troubles- so could using the wrong length ammo. If barrel is not marked with length of shell, would have a smith check the length of chamber with a gauge. 12 g shells were not always 2 3/4 or 3 inch.
If headspace and chamber length are ok, you can try polishing the chamber. Nothing more abrasive than several layers of cloth wrapped around a bore brush, a dab of J-B's Bore Shine, and a drill.