Here are a few things to look at.
1. Full choke generally isn't the best turkey choke. Most "pros" you see on the videos are shooting a minimum of extra full. You might want to get a couple of different turkey choke tubes and pattern them to see which is the best. I have a Browning BPS with the standard invector choke tubes. I shoot a .660 Rhino choke tube, which is a lot tighter than full.
2. I shoot at a Quaker Boy turkey target. This has a 10 inch circle around it. My minimum acceptable shell/choke combo will put 100 pellets in that circle with 5 pellets in the brain/spine at 40 yards. I like 40 yards because if it'll whack him at that distance, it'll get him if he's closer. This will give me some extra in case the bird moves when I pull the trigger. You may want to start with this and modify the distance to fit your needs. The gun I use now puts 212 in that circle with 9 in the brain/spine. You should also shoot a minimum of 3 targets and compare the pellet counts, because they will vary. Nothing hurts any worse than having that gobbler at 30 yards and the worse pattern that the shells will give you being the one you send downrange and cripple that bird.
3. I have found that the hi-velocity shells seem to pattern better than the standard 3", 2 oz. loads, or the 3-1/2" 2-1/4 oz. loads. It probably has a lot to do with fewer pellets fighting for the same space. The Winchester HV loads have done fairly well in every gun I've tried them in (4) and with all the different turkey tubes (too many to even guess). That is a good starting point.
4. The #6 gives a good pattern, but remember that the pellets tend to lose energy quicker than the larger #4 and #5. This can lead to crippling the birds at 40 yards instead of a clean kill. One way to get around this is to use hevi-shot or some other non-toxic load. This will give you the energy of the larger pellets with the pattern of the smaller ones. The downside to these shells is cost, but only you can determine if they are worth it.
Finally, don't believe everything that the "pros" are telling you. The camera can make a 20 yards shot look like it's 60 and there is always the "golden BB" rule where one pellet is what kills the bird at 55-60 yards.
Hope this helps you get started and gets you killing some birds.