I bought a Mossberg Accu choke XX-Full Turkey Only one I could find local. I tested it when I got it home. I Put it in my Winchester 1300 with 28" barrel and fired standing up @ 35 Yards and than I fired the same distance with the Modified choke that came with the gun. My target was 40"x 35" the shells I used were Challenger 3" BB Lead 1 5/8 o.z. with Max dram powder. There was a strong wind 50km/h blowing in my face if that matters. Here are ther resulting pics:
XX- Full turkey
Modified Choke
Challenger 3" BB Lead

 

looks like the choke is way too much for the big pellets. I would guess it is flat siding the pellets, and the wind would have made the fliers even worse.

Try the chokes with smaller shot. Not sure what you are wanting to hunt with BB's and this choke, but many of the new waterfowl loads (tungsten etc) tell you not to use a full choke...

Steel shot also tells you not to use full chokes, will blow the pattern, as it tries to compress too much...

Try shooting some bird shot - 2 3/4" at paper, and then move up to the 3", etc. play with it, and see what your gun likes.

Good luck

Ken