Have you shot the gun at 10 yards to determine the center of your pattern? If not, you may be patterning a fringe of your center density, ie. the majority of your shot is centered off target.
I'm confused about your hesitation on getting hevishot though. It costs $2/shell. At $7/box, a normal turkey round costs you $0.70. Instead of comparing it by the box, compare it by the shell. For roughly $1 more per shell, you can double your average payload. If you only shoot 3 or 4 shells per year, that's $8. Most people spend that each week on coffee.
Not lecturing or pushing your opinion, but personally if I were you I would 1) re-pattern your gun at a closer range to determine center of shot, and 2) try the hevishot. One box of shells is much cheaper than a choke, and you will need to pattern multiple brands from the new choke to boot.
Also, out of curiousity, did you ask a gunsmith/tech at Stoeger which ammunition they find patterns the best? Remember, those guys work/shoot those guns on a daily basis, and know which loads pattern best...