Two things.
First I totally emphasize with you.
I'm a doctoral student, and exercise is really my only outlet for stress and anxiety. If I was injured and could not train, I think I would go nuts.
That said, you need to look at this long term. I *assume* you want to continue to workout into old age, and use exercise and weight training as not only a mode to stay strong, healthy and look good, but also as a therapeutic means to reducing stress and anxiety.
If that is the case, then continuing to work through this injury and not rehabing it is like driving with the check engine light on and engine knock. Sure, you can probably squeeze another 5 or 10,000 miles out of the car, but eventually it is going to break down and cost you 5-10x as much and as long to fix.
Unfortunately, you only have one body. So try to imagine how you would feel if you need shoulder surgery and then cannot lift for an extended period of time.
Get yourself some bands. You can buy them for 5 bucks at ocean state job lot or order them online for similar
Br