I think you should lay off any direct pressing right now. To better assess on what sort of injury you have - because you might not be injured at all, I'd like you to do something.
First, I want you to do Overhead Shrugs with a barbell. It's a shoulder prehab exercise. Tell me how these go for you. Do 3 sets of 10-12 reps. This should be light and easy. You aren't supposed to go balls to the wall or even look for a challenge while doing these. They're an assessment test.
Next, do some facepulls. 3-4 sets of 10-12 reps. Again: these should be easy. I'll give you videos so you know what I am talking about.
Then, do some push-ups. How do these feel?
What I suspect is just some temporary wear and tear which you're experiencing. Take some time off from the pressing, do the shoulder prehab exercises for 2-3 weeks at the most and you'll get better.
The drawback of shoulder issues is that you have to work very hard on your back exercises. Doing just one row exercise won't help you long term.
You should also consider seeing a doctor. I remember feeling like you a couple of years ago and these were symptoms of people who have genetically loose shoulder joints. This means it's easy to have shoulder dislocations. I wish I had worked on my prehab and rehab at the time but I was ignorant of these details and didn't see a doctor till I was rushed into the hospital with a dislocated shoulder. So please consider meeting a professional.
If you do have shoulder issues, it's fine. I can help you get back on your feet once the doctor deems you having the problem and stabilizing yourself from injury's way.
Also, you mentioned having the problem while doing rows and stuff. I still don't think you have shoulder injuries per se but I think you should lay off the barbell rows and do something which is more efficient at what you're wanting: a strong upper back. Do some dumbbell rows and cable rows.
So yeah: first do the prehab exercises I listed out and drop all pressing exercises (save the push-ups) and tell us how they felt. You might be fine in just a week so don't think you are compelled to lay off pressing for 2-3 weeks. You're not. But 2-3 weeks is a good range to gauge whether a problem exists or not. BUT, I think you should see a doctor nonetheless. We're just a bunch of people typing away on a forum...a doctor will be FAR more suitable in assessing you.
Overhead Shrugs:
YouTube - DeFrancosGym.com - Exercise Index: Barbell Overhead shrugs
Facepulls:
YouTube - DieselCrew.com - Shoulder Rehab - Face Pulls
Dumbbell Rows:
YouTube - M15 W3 Squat Day.wmv (start at 1:50)