Although it doesn't seem fair (and it isn't), the first thing you should try it buying another wedge.
With this design, if the wedge isn't wide enough it won't drive the barrel back to the correct position.
If you look at the pictures, you will see that there in a gap between the barrel and the frame (at the bottom).
This area shouldn't have a gap at all. It should be tight.
If the barrel was tight against the frame there might still be a small gap between the cylinder and the barrel but this isn't all bad.
If fact, it will keep the cylinder from locking up due to fouling.
Yes, there will be a lot of flash coming out of this area when the gun is fired but, as long as the chamber and the barrel are lined up and the cylinder is locked by the cylinder stop when the gun fires, IMO it is safe to shoot.
Until the gap at the bottom between the barrel and the frame is eliminated though, I would not shoot the gun.
I don't know if Tickfood can measure his wedge but as a reference I just measured 4 of my guns wedges. The width is tapered with the small dimension on the right hand end and the large dimension on the left end (screw side of the gun).
These dimensions give the small end, the large end, the length and the thickness in that order.
Uberti 1860 .496, .525, 1.080, .138
Pietta 1851 .478, .523, 1.075, .138
Pietta 1851 .481, .519, 1.076, .138
Pietta 1851 .480, .527, 1.074, .136
If Tickfood doesn't have a micrometer or a vernier caliper a good ruler with 1/32 inch markings could be used. If the width of the narrow end is a little less than 1/2 inch and the large end is 1/32 larger than 1/2 inch then the width isn't the problem so buying a new wedge wouldn't solve the problem. If the large end is 1/2 inch or less, I'm betting that buying a new wedge would fix the gun.
These dimensions also tell me that the Dixie Gunworks Catalog is correct. The same wedge will work in the 1851 and the 1860. Their part number by the way is #PP 1808 and a few years ago they wanted about $7 for one.
Not that they are recommended as being well made but many of the "Colt" style pistols that were made in the Confederate States had much larger cylinder gaps (not frame gaps) than the ones on this gun and they were functional.