Can I suggest you go see a shotgun fitter? Commercially made shotguns sold in the US are typically made to fit a male, about 5'8" tall, right handed and right master eye, and about 180 pounds. If you deviate from that, you probably need a fitter to make some changes to your shotgun.
I am 6'4", weigh 235 and need really long sleeves on my shirts, and I typically broke about 15 of 25 in Skeet, until I saw a fitter. With that, and a little advice, I suddenly typically broke 22, maybe more.
The most important of the advice from the fitter was that if you are right eyed and right handed, you'll aim your shotgun with your left hand. He favored grasping the front wood out near the front, and do so extending the left index finger down the barrels. He wanted me to focus on the bird, but look ahead of it by the lead I needed, and the shotgun would follow my eye. When in doubt about the necessary lead for a bird, use the longer estimate, because the shot cloud is actually a teardrop, with a long tail, and if you err on the long side the tail will still nail the bird.
Regarding what shotshell hulls are best, the answer used to be easy: Winchester AA. Then along came the various compression formed Remingtons, which were as good. Of late, I have been using a lot of the European straight wall hulls, and their life is excellent. If you're laying in a lot of empties, be aware that the "Euro-trash" empties use a slightly larger primer than most US made ammo. Most US primers will be loose in the European cases, and may drop out. Conversely, I have used a lot of European spec shotgun primers in Winchester AA, Remington & other US spec hulls, and they work fine, but wreck the hull for future use with most US primers. You can find lots of discussions regarding these matters on the Shotgun World discussion forum.