If you cut the shell back, you lose the folded over lip. This lip is important for several reasons:
A wad is compressed a bit on the powder. This compression is important to reliable ignition. If the lip is cut off, the wad will relax a bit.
On recoil, the lip holds the wad and shot in the case. Otherwise it will tend to pop out the top.
In the case of a wax slug, the slug is as wide as the walls of the shell, so the folded over lip holds the slug in place, so it can't pop out and block the bore.
If I had a 3" gun that would not feed 3" shells, I would either switch to 2 3/4" shells or look closely at it to figure out what is happening. The first thing I would do is ask my fellow CGNrs with the same gun if theirs will feed 3" shells reliably. If "yes", then I would work on the gun. Could be something as simple as a bent follower.