I started doing the olympic lifts at 34, unfortunately self taught. I say unfortunately because it's harder to unlearn a bad technique then to learn a good technique from scratch. I found it much easier to power snatch everything, in fact I couldn't do a full snatch to start with (no idea how plus inadequate mobility) and it has taken a long time for me to stop reverting to power snatching when things get heavy (heavy being relative for me).
99.9% of my snatch misses are to the front, and there is a natural tendency to try to 'save' a lift. I too used to 'tweak' my shoulder doing this, at least once bad enough that I went to the physiotherapist about it. In my case it was a rotator cuff problem, although the pain was not in the back of the shoulder. This was fixed with some specific rotator cuff work, and as my mobility and technique have gradually improved I no longer have this problem.
Once your technique improves (and mobility, if that is a problem) you will find that your power snatch should be less than you could 'full' snatch. This took a long time for me! Perhaps I am a slow learner. Anyway, if you want to become a good snatcher than I agree don't give up on the 'full' snatch entirely.
Not sure if this helps you but it sounded kinda familiar so I thought I would drop in my 2c.