A timely topic. Just last week I completed surgery for a torn supraspinatus tendon, torn tendon in the long head of the bicep, and some labrum damage.
It all started a couple of years ago when bench pressing using a machine (never again!). My immediate response was to avoid the machines and the pain went away--for a while. But it kept returning on different pushing exercises. It gradually forced me to trim back the number and kind of exercises I could do. I finally got to a point where I couldn't even do a darn push-up; couldn't ride a mountain bike without a sudden bump causing searing pain.
I tried PT. But relief there was only temporary.
I finally bit the bullet when it was explained to me that: 1) Torn tendons rarely heal on their own and, 2) I was quite likely only going to cause further damage to my cuff if I kept going. This is the overcompensation issue you mentioned, OP.
Obviously OP, if you can get relief from PT alone, that's best. As my own PT put it, you can try to strengthen some of the adjacent tissues. But be on guard for further injury like you're noticing.
If you do go for surgery, look for a doctor who does hundreds of these per year. That's what you want--the kind of guy who focuses on shoulders and does them all week long.