It sometimes happens that a choke tube can over-contrict the pattern and actually make things worse. It happened to me once with a BPS I owned: even the Full choke produced wider patterns than the Modified when using copper-plated 3" sixes and fives. And commercial turkey chokes were a joke in this particular gun. So, just because a choke says Full or Extra Full doesn't necessarily mean your patterns will always tighten up. You just have to experiment and find out. I'm more worried frankly about the "high and left" shooting, which MAY not correct; try different loads and see.
The late Don Zutz, one of the best shotgun writers I've read, once described shotgun chokes as being like an adjustable spray nozzle on a garden hose. You can twist the nozzle and watch the water stream get tighter and tighter...then you reach a point where further twisting only makes the spray pattern worse. The same holds true with your chokes. Your Mossberg, being the individual that it is, may prefer a more open choke to produce its tightest patterns.
"For every complex problem there is an answer that is clear, simple.....and wrong." H. L. Mencken