What is the bore diameter( Land to land) and what is the groove diameter? Saying the rifling is fairly shallow tells us next to nothing.
I would recommend doing to things differently than you now do with this gun.
1. Clean between each shot. use a little spit on a cleaning patch first, running the cleaning patch down the barrel in 6 inch stages, and then reversing and pulling it back up. That loosens the crud, seats the patch in the patch jag, and really cleans the grooves much better, or more thoroughly. The damp part of the patch in front of the jag will dissolve and then soak up the cruk on the breechface. If you gun as a smaller powder chamber, you will need to clean it with a separate patch, and jag. No getting around that. You can use a small bore bursh sized to fit the powder chamber with a cloth cleaning patch in front of it. The wires of the brush will grab the patch and let you pull both back out. Just make sure you turn the brush in the direction of the turn of the rifling. Most are right hand twist, so that turning the brush to the right also tightens the brush on the end of your ramrod. To break the grip of the brush, simply turn the entire rod to the right( clockwise) to lay the bristles over, and that will allow you to pull the brush back out the barrel, holding the cleaning patch in place.
When you are done with the wet patch(s), use a dry patch or two to dry the entire barrel and pull the rest of the crud out the barrel. A second dry patch may be needed in very damp conditions.
In the wintertime, when temperatures are below freezing, use alcohol, rather than spit to dampen your cleaning patch. The alcohol will evaporate quickly.
2. Lube a cleaning patch with wonderlube, or your other lube compounds, and lube the barrel after seating your ball. This will leave lube to soften powder residue as the ball passes, and the smoke and crud attach to the sides of the barrel. That makes cleaning much easier, but also helps keep the crud from locking onto the grooves in the first place. You will get a few FPS more velocity with a lubed barrel, and the Standard Deviation in Velocity will drop. Those are three good reasons to use this extra step.
Best of wishes.