I don't have the rep here yet to upvote or +1 a comment re: @jqning 's answer so instead; I offer this "I second what he said about Townie."
I suggest Townie (w/ admitted/disclosed bias) because it is what I ride. I don't have a bad shoulder but I do have 1 artificial knee, 1 artificial ankle and nerve-damage in my right hand; all of which limit mobility. The model I chose is a 2011 18-spd 2200d electra. It suits my physical handicaps more-than adequately; being a "bike-person" few of my materiel possessions out-rank it. it would be almost impossible to replace; if I had to I'd go with their current model; which would not be optimal for me...but for someone with shoulder problems -- it would.
the problem with your current bike is the stance. you're pitched too far above and in-front-of the bike's center of gravity.
for your shoulder, you want to be hanging-back...behind it--like on a cruiser-style frame--- where the handles are elongated and reach back (even downward a bit) to get to your hands. **This needs to be done comfortably, safely and most of all, the bike still needs exceptional vel. & accel. (because it's a commuter) and primarily, **it needs to be fun to ride.****
check out their specs .... test drive an Electra -- the current models comes in 7speed (1 up front, the 1st gear in back is extra-extra low-low). try similarly-designed-bikes too. if I could list other brands/models I would...well Schwinn and the brand you have now likely produce a similar style - many of the larger brands have knock-offs; some =ly good; some (possibly) better; depends on your budget. This style can easily (prob. w/o a custom/after-mkt handlebar, seat, etc) be adjusted to lower your center of gravity so when ur ON the bike, it takes all (an extremely significant amount of...)the pressure off your shoulders and redirects/displaces it to parts your body that can bear it.
bottom line - in your question you said "...I want to give up riding." even though I take this a bit out-of-context...there it is; in the lead of your question - if you really did? you would not be on SE asking other OCD socio-paths what to do.
The bike is the problem - scrap it:
- try a bunch of new bikes
- new designs
- new brands
Change "I want to give up" to "I should change bikes- don't give up."
Hope I don't sound like an advert-- the company no longer manufactures the 2200d - which to me, is a neg on their business strategy. in all fairness this statement should bring some objectivity back into my inherently biased answer.
If you LIKE riding; please don't quit. it beats walking, it's healthier and safer on your bones & joints than jogging and is comparatively FREE as opposed to a combustion engine, electric or hybrid ANYTHING. ALSO; the energy spent makes YOU healthier, as opposed to the spent energy of other vehicles; which mainly damage pocketbooks, the economy and the air.
Stay up, Tapper