Ringing the barrels only tells you if the ribs are soldered the entire length of the barrels. I've been shooting them from around 2003 or 4. I put on my first SxS Shoot in 2005 and many guys show up with Damascus barreled SxSs. Most shoot nitro powders, and no one has blown up a barrel. This is not to say all Damascus guns are safe, but Sherman Bell in two different test tried to blow up 40 "wall hangers" with Remington proof loads of 18,200psi. A normal load is max at 12,000psi. None of them blew. He then took a Parker and kept increasing the powder and at 30,000psi it finally blew. Next load at 32,000psi and the other barrel went. The British always have and still do proof Damascus barreled shotguns, rifles, and pistols. They, unlike us, see nothing wrong with the use of Damascus for barrel steel.
I try to show 100+ year old guns a little respect. I use lighter loads because of the wood - not wanting to see it crack. And most the old SxSs have a lot of drop in the stock. This will cause the barrels to go up along with the comb of the stock. This means your cheek bone is gonna get whacked. BP shotgun loads are in the 7 to 8000psi range, so picking your load around there will help. Also, light [ 1oz or less, I like 7/8 or 3/4oz in the 12ga ] shot charges. It's hard to find these kind of loads unless you reload or know someone who does. As Hawg said, 65 to 75 grs of BP will work with 1oz of shot. Get one of the old adjustable BP measures that can be used for both powder and shot. Using one for nitro powders is where back in the day many guns were blown up. Guys were use to using equal amounts of powder and shot, and that didn't hold true for the new nitro powders. And that's one of the reasons Damascus guns got a bad reputation. Anyways, if shooting BP use fiber wads, not plastic. They'll leave a mess in the barrel. Plastic shells are ok. BPI or Precision Reloading have adjustable measures and fiber wads. Circle Fly, who I believe makes the fiber wads for everyone, has a web site with a chart explaining all about fiber wads. With BP I've shot anywhere from 70grs of 3F and 1oz of shot to 100grs of 2F and a 1 1/8os of shot. BP is very forgiving. Nitro is not. You can find light nitro loads in the Hodgdon or Alliant web sites. I use mostly Alliant and for 3/4oz loads in a Remington hull and Red Dot powder, Claybuster wads show 15 to 17grs or 15.3 to 17.3 in a Federal hull. All loads are 5500 to 6350psi, which is a light pressure load, light shot payload. There are many more, you'll just have to do a little research.
I now have about a dozen SxSs, all but three with Damascus barrels and have been shooting one or the other three times a week over at the club near my house. No one can answer your question if yours is safe. And not a gunsmith who doesn't know something about Damascus guns. If he doesn't own and shoot one, he doesn't know. They're just gonna CTA in case of a lawsuit and say don't shoot it. You can look at the gun to see if it was taken care of. Is it still tight on face [ no wiggles side to side, up or down , when closed ]. Is the bore smooth ? A little pitting is ok because most were shot in the BP to nitro days. Is there somewhere around .100 thickness at the breech end, .040 or more at the muzzle end ? Do the barrels ring ? [ stick your finger out and put the hooked underlug on your finger so the barrels are hanging on it. Tap the barrels up and down in a bunch of different spots with a quarter to see if they ring. Maybe they won't in every spot, but should ring. ] These are a couple of test I make when buying any SxS. Hoped I've helped a little.