George W. Tryon had several contracts for military arms in the pre-Civil War period, including a contract for the Model 1841 rifle. Whether there is any connection with the manufacturer of your shotgun, I don't know.
The so-called Damascus or "twist" barrels were made by twisting thin bars of iron and steel together, then heating them white hot and wrapping them around an iron rod, called a "mandrel", in a spiral ("barber pole") pattern. While the bars were still hot, their edges were welded together by hammering them. The breech end was either made up of thicker bars or of more than one layer.
When the wrapping and welding finally produced a tube, it was filed smooth and reamed out to make a barrel. While adequately strong when new for the old black powder, most folks recommend that guns with Damascus barrels not be fired. In the first place, even modern black powder is stronger than the old type and in many cases, rust and corrosion has gotten into the tiny cracks in the welded barrel. I have seen Damascus barrels that looked more like lace than solid metal.
The spiral pattern can be seen in most Damascus barrels, and can be quite beautiful. In fact, for many years after solid steel barrels came into use, many customers preferred Damascus barrels for their appearance, in spite of their being weaker and more expensive. Some makers even faked Damascus by making a spiral pattern on solid barrels.
I have no idea what you were told about the barrels not being "wrapped", as that is the way Damascus barrels were made; I suspect that the person who said that was either trying to sell the gun or was in error. It is possible that the barrels of that gun are fake Damascus, but unless you have it checked by an expert, I would assume that the marking is right, and indicates actual Damascus.
The fact that the gun has a serial number indicates it may have been of good quality, but I still recommend it be treated as a collector's item and not fired.
Jim