Hi!!!!
Well, I can give you some tips as I'm in a similar situation. I'm a manual chair user due to Generalized Dystonia and I've developed the exact same issue with my right shoulder. If it flares bad I can't lift my arm up above my head. My shoulder usually hurts or is just sore. I had PT once for it and they used an ultrasound and ice which did help, but obviously pushing a chair, transferring, pulling open doors, manuevering carts and garbage cans make it worse and worse.
Power chairs have their own problems. I have a very nice one, but the batteries died and I haven't got them replaced (no medical insurance) so I'm back to a full time manual chair user.
What I got told with my shoulder is to stop or greatly reduce pushing the chair, use ice and Advil. Mostly I have not stopped pushing the chair, use heat and Tylenol. lol.
Those power assist wheels for manual chairs are first off expensive, second I had looked into them myself for the exact same reason and discovered that users gave them poor reviews for various reasons. Because no one seemed happy with them I let it go and didn't get them.
Do you have a light chair? My current one is not a light chair and is more difficult and heavy to push. At some point I will buy a lighter sports chair that is much easier. That would be one thing that would help you. Switching between a power chair for long outings and a manual will also help you out. Honestly, I've dealt with the shoulder injury for years now and it hasn't gone away. There isn't really a way for me to save my shoulder. I have to get around. Even steering my van (with my right arm of course) makes it hurt. There isn't really much you can do.
So I would say things that help me are:
get a lightweight sports chair that is easier to push
rotate between a power and manual chair (if you are able to do other exercises that are low impact on your shoulder that would be good to keep your weight down)
use ice and Advil when it hurts
As for the dog, yes they can help you. I had a pulling dog for 10 years. I specifically got this dog for pulling. When you apply for dogs make sure to stress your interest in pulling. My Lab was big, strong and very driven and his strongest task was wheelchair pulling. I had a sports chair at them time and am a normal weight, he almost weighed as much as I did. He would pull my chair anywhere at anytime and our speed was usually a trot.
The harness I used had handles on the side that you would hang onto with (thankfully) my left arm and tell him Forward. It got to the point that no command was needed, including Left and Right. We were completely in tune with eachother after about a year.
This task also freed up my other arm to do other things like hanging onto stuff, talking on a phone, holding a drink, etc. It made me move faster, be more independent and save my shoulders.
After he retired I tended more toward a power chair. As a result I did not pursue another pulling dog. My current dog is not a hardcore puller and doesn't have the body frame or drive for it anyway. He can and does pull me on flat surfaces though, but nothing compared to my pulling dog.
So, you can research the power assist wheels, just make sure to look at the reviews for any given model. Other than that you just have to do your best to preserve your shoulder and hopefully you can get a dog that can help you with it.
If you have anymore questions I would be glad to try and help you out the best way I can

Welcome again I'm glad to see you back!