Kyboy... he indicated he had a "37a"... so it's probably a single shot .410 bore Winchester Model 37A shotgun.
As to the 3" vs. the 2½" shells... the 3 inch .410 shell has a 3/4 oz. "payload" whereas the 2½-inch shell has only a ½ oz. of shot, so under normal conditions, the 3-inch shell would have half again more shot in the pattern... and should have a 50% increase in pattern density... meaning it should have a 50% greater shot density than a 2½ shell in the pattern.
However, I suppose it is possible that a very old Model 37A might be chambered for 2½-inch shells ONLY. The chamber for a 2½ inch shell would measure 2-3/4 inches long. The chamber for a 3-inch shell will measure about 3¼ inches long.
The final 1/4-inch of the chamber is for the rolled crimp of the shell to open into. Is the size of the shells that shotgun will shoot indicated on the outside of the barrel near the receiver on the left side of the barrel? If it is designed to accept 3-inch shells, the barrel markings will tell you so. If it is marked "2-3/4 inch only", then you should not shoot 3-inch shells in the gun.
If it is marked for "2½ inch shells only", then that is the reason "holes" might occur in your pattern when you're firing 3-inch shells in too short a chamber. The rolled crimp of a 3-inch shell doesn't have enough room to open up and lay flat and get out of the way of the shot coming out of the shell.
By interfering with the shot's ability to get out of the shell, down the barrel and through the gun's choke without interference, this could be the cause of an uneven pattern when using the longer shotshells.
Strength & Honor...
Ron T.