I have a Rem 870 SPS-T 12 guage and all I heard was great things about the Hevi-shot Hevi 13's and so I went and bought some. They patterened the worst. I found that the Federal flight control and then the Winchester Supreme's patterned much, much better. I am getting between 8 to 12 hits in the brain / vertebrea with the Federals and the Winchesters performed just a tad bit worse than the Federals. I noticed that you are using 3" shells...my results were with 3.5" #5 shot using a Primos Jellyhead.

As stated earlier...make sure you are shooting dead on (sights accurate). Check this by taking cheaper target loads in 2 3/4" and stand about 15 yards away with your turkey choke and blow the center of the bulls-eye out. Stand really close will let you know if your pattern is off to the left / right / up / down. Make the needed adjustments and then try your turkey loads from 20 yards.