I can commiserate with the op. I have shoulder impingements in both shoulders from years of injury. Physical therapy was only mildly successful and avoiding shoulder surgery is a priority, especially since nothing is torn, however, they bother me every day...which is why I stopped playing drums for a number of years and went back to guitar, but I'm back to drums too..can't stay away. My kit is smaller and easier to reach, and I've worked at changing technique from the standard arm thrashing of a rock drummer to more wrist and elbow. That's been helpful.
As far a guitars go, SG's are lighter and similar to an LP, however I never liked the neck diving the moment I let go of it...and the ones I had, had necks like baseball bats. Try a chambered LP or LP copy, or any one of a zillion bolt on models made of alder or basswood.