Try rotating your right hip back more in the backswing. Almost feel as though your right hip pocket goes behind you (it won't but that's the feel). This will allow you a deeper turn and more room to swing down more shallow and on the inside.
A good drill is to take a dining room chair and put it against your right hip to make sure you don't sway. Done correctly your right hip should actually come off the chair in the backswing as you rotate your right hip back behind you. If you bump into the chair you've swayed.
Again, the whole idea is to get you more room to swing more shallow. Make sure you start the swing with your arms and not your shoulders so they can find the slot easier. A swing thought of keeping your back to the target longer isn't a bad thing also.
Give it a try. Get that right butt cheek going behind you and to the left. Your ball striking should improve as well.
Red Sox fan as well. Transplanted Cape Codder now living in Florida.