If your shoulders are impinging, you need more upper back and rear delts. Build a more stable foundation. Something like you have outlined above would be pretty good. 1 or 2 upper back moves (Cuban rotations, Y-Raises, face pulls, shrugs on an incline bench) and 1 or 2 rear delt moves (rear felt flies, rear felt swings, rear felt extensions on a cable). Really learn how to contract and squeeze the muscles back there. Shrug and squeeze and tilt your scapula. Avoid pain, progress slowly.
Once your shoulders are more stable and you can control your scapula, move to Phase 2 and add 1 side delt exercises. Side delt raises with dumbbells or on a cable worked for me. You may have to stare at your shoulder in the mirror or put your off hand on your middle delt to make sure it’s working. Low weights, maybe even short ROM at first, until you can make that side raise motion pain free. Don’t allow the front delt to take over and roll forward. Avoid pain, progress slowly. Continue to progress upper back and rear delts while you learn to use side delts.
After that, add in the Bradford Press with super light weights, like a broom stick at first. This silly move can help you rebuild the pressing motion, but with an emphasis on side delts. Progress slowly. Learn to get the stick to your collar bones without allowing front delts to completely take over.
If you make it that far you should be ready to overhead press the 15 pound dumbbells next year.
Here’s a picture of the muscles that hold your scapula
2 upper back motions
Rear delt swings to rear
To the side
Future Bradford Press.
More info on scaps and upper back.
Article about training shoulders without free weights