I'd go with the cork wads and see what happens as far to accuracy.
If they only move forward sometimes, then the cork should work.
After being fouled, there should be no movement, but I'd still use the wad.
My .58 Kodiak shot well (parallel) with 110 2F GOEX., a .020" denim patch
and .574" pure lead ball. It also shot well enough (3" to 4" groups at 100 meters)
with a .562" WW ball and the same patch, same powder charge.
Some guns like Minnies, while others refuse to shoot them.