[rogn]

Since it rotates freely without cases in place and worsens with use then fouling is certainly part of the picture. The type of ammo and /or cartridge may be part of the picture. If youre using a low intensity cartridge and your revolver has headspace that is on the long side of specifications the primers may "set back". They will inturn drag on the recoil shield. As you fire more rounds the fouling in the chambers may keep the cases pushed back against the recoil shield. If you coat the case head and primer with magic marker and examine them after firing the cylinder full. On observation of the fired cases the primers will protrude above the case head. This may work in various repeaters but not in most revolvers. One important thing to consider also is that the little "hand" thingy is what rotates the cylinder. They tend to be kinda spindly so overloading it can affect timing. Only a very small dimensional change can make a profound difference.
Knowing what cartridge and load youre using is of interest, regardless. If youre using cricket fart loads then primer set back is likely the cause of your difficulty. In that case the cure may be as simple as using normal powered loads. If the recoil shield is excessively "ramped" the fix is going to fall into factory work.