I have some back issues and shoulder but I still do some I just don't do exercise that put pressure on back like no squats but I do leg extensions and leg curls for legs and lat pull down for back. Not saying my back is worse than yours but am missing disk in spine and it does hurt but still do something
I started lifting weight 6 months after the shoulder dislocation. But I still did not try push up, which could have hurted my shoulder. It past 7 months since the shoulder dislocation and It's been more than a month since I started working with dumbells. Should I try the push up 2 days later?