Sounds like rings may help, especially if you have no pain when doing push-ups.
I don't need a partner for set-up, and adding weight isn't very hard. I bought two 23 lbs chains that I wrap in an X formation (over and under my arms, if that makes sense), which allows for full scapular mobility. They have large carabiners, so I can attach extra plates on them (as long as the plates are the kind that you can stick your fingers into, as opposed to the ones that only have a hole in the middle). If you have the latter, you can add a smaller chain to the larger chain, and simply slide it through the middle of the plate. The chains will hang low, so it's best to have your feet elevated on a bench to avoid them touching the ground (unless you want to train the upper portion of the lift). I go down until my arms are at a 90 degree angle (just like a floor press).
As a side note, you may want to purchase rings anyways. They will also help tremendously with dips. I found bar dips easy, then tried ring dips, and at first, I could only do 3 or 4 (and that was without added weight). You catch up quickly though, but again, my shoulders love them (especially since I read Dr. Joel Seedman's article in December "Dips: You're doing them wrong"...he advocates the hollow style, which will further protect your shoulders).
So, my advice to you: buy the rings, and at least try hollow dips (which will work your chest and triceps). For shoulders, you can rely on overhead pressing since it doesn't hurt. I'm not an expert on overhead pressing, but I don't think they will work your sternal head (lower chest) very much, but the dips will in the hollow position. Then there's ring pull-ups and ring inverted rows, both of which provide an extra challenge compared to traditional versions (note: because of the angle, inverted rows work better for me with a backpack - chains slide down and choke me).
Hope this helps