I love finding a good solid antique and bringing it back to service.
I call those, "having good bones", and
@Relic shooter, "worthy".
When looking at an antique shotgun, I'm really picky about barrel condition.
Ribs- upper and lower- Look carefully for any separation from the barrels. Acidic flux was sometime used. Over the years corrosion can form on the steel beneath the solder joint. This will cause the joint itself to weaken and fail. Look for any evidence of a gap between the rib and barrel. Sometimes you can see the corrosion (rust) peeking out from a seam or a hole in the solder. Any repairs to fix this will completely change the patina and look of an antique. Sometimes you will see oxidation, too. This is a white chalky layer that forms on lead based solder. Oxidation is not as much of a concern as corrosion.
Barrel delamination- Look up how "twist steel" barrels are made. Most of the older shotguns are of this construction. These layers, or coils, can sometimes delaminate due to conditions. I use a nickel to bounce all along the ribs to check for rib separation, and all over the barrels to check for delamination. If you land on a problem area, your nickel will give a thud instead of a dink. I walk away from anything that thuds, no matter how much I like the gun.
Bore condition- That delamination is caused by rust and corrosion eating pits into the bore and then creeping into the layers of the twist. For me, If the bores are any more than very lightly pitted on these barrels, I choose to pass. Surface rust, though, can be polished off.
If you are going to be shooting round balls out of antique barrels, I would make sure that they are
not tightly patched. I don't use modern plastic shot cups in an antique barrel, either. I stick with lubed felt wads and overshot cards.
I also keep in mind that I am shooting a gun that is 150+ years old.
Good luck with your search!
With many satisfied shooters, its hard to go wrong with a Pietta or Pedersoli double.
Take your time if you decide to purchase an antique, foremost for safety, but also for value and resale.