I am pretty sure I have shoulder tendinitis.
I say this because I had wrist tendinitis before and this feels the same, except in my shoulders where the joint connects to the collarbone.
I first developed it after a SD cycle where I was doing alot of heavy flat dumbell presses. It is quite annoying to say the least, especially because any chest exercise I do seems to aggravate it (I don’t do any direct shoulder work.) Chest is very important to me but I cannot seem to exercise it without activating the shoulders whish is very frustrating. Anyone have some tips on which exercises or techniques I can use to continue exercises chest with minimal use of shoulders?
If I were you, I’d go visit a doctor about the tendonitis and make sure that’s what it is, and get some help with treatment (drugs, physical therapy, at-home exercises, etc.)
I tore my AC in my shoulder over a year ago now. Put off treatment for 5 months and my shoulder STILL isn’t right. The only thing that helped was resting it (no swimming or weightlifting or pushups) and then 3 months of PT just to get things working right, and then SLOWLY building up my workout so that I could do the same things with my right arm as my left.
I got it back to “normal” but then took a month off of working out to move into a new house, and it went back to not working right again. The only thing that seems to help it is consistant working out, keeping the surrounding muscles strong.
So yeah, shoulder injuries can severely mess up your workout routine and not getting proper treatment ASAP can make things much worse for a very long time. Go see someone “in the know” about these sort of things before you’re screwed for life.
My kneejerk reaction is you probably won’t be able to find a chest exercise that won’t also aggravate your shoulders. Just the nature of the best since your shoulders are largely connected to your chest. If it’s not directly involved with force generation, it’ll be directly involved in stabilization.
You should be asking this question to a doctor with experience in physical rehab/sports therapy instead of a bunch of clueless schmoes on a message board. 
I had something similar happen to me several years ago. I couldn’t do flat benchpresses all of sudden because of shoulder pain. Strangely, though, I found that I could use the Hammer Strength machines just fine and I’ve been doing that ever since. I try to use freee weights as much as possible, but the Hammer Strength machines are a good compromise. If your gym has them, try them out and see how you feel.
You might find a ‘Smith’ machine to be helpful, and decline presses might be easier on your shoulders, provided a DR. gives you the greenlight.
Smith machine is a freee wieght style barbell (plates and all) on a guide, which prevents shoulders from rotating during stabilization. This could really help, as the weight cannot rock back and forth, only be pressed up/down.
Decline presses also take some, not all, pressure off the shoulders, as the weight won’t feel like it is rolling back over your head (which seems like the most strenuous things for sore shoulders).
What they said about seeing a doctor. Work on your upper back now.