It isn't so much about the gun as it is your technique. Most any gun that fits you will work as I have seen many young shooters with single barrel guns smoke targets like crazy. Assuming the gun fits you, there is NO shotgun made that won't break the targets if YOU are pointing the gun in the right place when it goes off. You need to FORCE yourself to not lift your head, even if you miss a couple of birds in the process. If you concentrate on holding your head down and forget about seeing the birds breaking you will see an improvement quickly. There is no need to lift your head to see if you broke the bird, the peices will be in the air long enough to see after the shot. The only thing more guaranteed to help you miss more birds is stopping your swing.
I had found during my many years of Skeet, Trap and S/Clays that you reach plateaus in your shooting scores. You will hit a point that just seems as though you will never get better or improve your scores. With a sound shooting basis and following the ("proven rules") you will improve, maybe only a bird or two at a time, but you will improve with dedicated practice.
I learned a helpful hint from a great shooter "D. Lee Braun" many years ago.. Empty your mind of EVERYTHING other than hitting this single bird, not the next one or the one you just missed or hit,, just this one. If you had a fight with your wife that day, you need to forget it for 3 seconds if you are going to hit the bird. you need to concentrate intensly on this one bird for just a couple of seconds.
It looks like you have a GOOD coach, listen to him,, he has proven what works by his scores. Everybody will give you advice, but, as I found out years ago, I listened only to the ones that could kick my butt in the game, not the ones I outshot every time.
I will say that a pump gun is not the best to use for the doubles as it is a bit slow and that is why auto's and O/U's are so popular.. fast second shot without doing anything. As far as a 20 guage. it will work great. I shoot a Browning 4 barrel set and LOVE to shoot the 28 guage the most, followed by the .410. Even a .410 will break the birds at skeet, it may not powder them,, but it breaks them and that is what counts.
My personal opinion as far as a gun goes would be as posted above a Rem. 1100/1147, Beretta ( I use both at times) or a reliable O/U. Check out good used guns too. There are almost always a gun for sale at the Clubs. This is because there must be something we inhale from in the dust from the targets that somehow drives men to thinking they need a new gun. Best of luck in you new found sport