More exercise also necessitates more sleep, which can be hard to do when there isn't enough of time.
If you're experiencing exercise-induced injuries immediately during a workout (and if you have a budget for this), it might be good to talk to someone about what you're trying to do. Some gym plans include a bit of consultation time with a staff trainer, and they can probably help you work around whatever the problem is, sometimes simple adjustments can help (particularly anything involving preexisting shoulder damage, most of the commonly-recommended shoulder exercises have substitutes that are less risky for already-damaged shoulders).
1. https://en.wikipedia.org/wiki/Delayed_onset_muscle_soreness
Feeling sore in the muscles is normal, feeling joint pain is not.
It's true you need more rest/sleep but I found you can work around it if you don't care too about putting on muscle and thus work out less frequently and with less intensity, just for the health benefits.