Some bicycle geometries are more stable than others, for example they can be ridden hands-free.
Stability is related to wheelbase and something called the mechanical trail.
Maybe bicycle that is stable enough for hands-free riding would be beneficial in your situation.
You can easily try out a few bicycles and see which ones rides best hands-free, and then choose that bike as a starting point for the modifications described by other answers here.
Also, having an internal hub give you another braking option.
A bicycle that is more stable means that it requires less frequent corrections to get it back on course. The more stable the bicycle is, the lower the wobble frequency (I don't know the technical term for this, but on a fast bike, the rider has to immediately correct any deviation, a touring bike in contrast is more sluggish and requires fewer and slower corrections).
Roadbikes are made to steer very quickly, as this is a requirement for racing.
Having a more sluggish bike (like a touring bike), might make it easier for you to hang on to the handlebar with one arm, as you will need less control input overall to make it go straight.
On a short wheelbase bicycle, you'll have to constantly correct in order not do trip over.
So try out a classic long wheelbase touring bicycle with a stable steering geometry. You will be able to sit more upright with less pressure on your arms and back, I like their classical square diamond frames.
Another alternative might be a recumbent, they come in many different shapes and most have a massive seat that gives you a positive mechanical connection to the bicycle, so you can lean back and don't need to rely on your arms as much.
There are also no-hands recumbents, here is a discussion about them http://www.bentrideronline.com/messageboard/archive/index.php?t-28633.html
hope your shoulder gets better soon!