I hate to be a scaremonger but, yeh, i've had a similar pain, but it was only in one arm / shoulder. I ignored it and ended up with a partial tear and impingement syndrome. Don't know which came first, though, probably the impimgement.
However, before the pain, my gains stopped pretty drastically. I just hammered away thinking it must be normal. It started to just feel 'weak' at first. Difficult to describe, but just a real feeling of weakness. Sounds like that feeling of weakness could be what you are describing. That developed into an ache I would get after sets and after the work out. Got worse and worse and I ignorantly just kept going for it. Then it went, similar to what you described, but the pain didn't go away.
It's a pretty long story to describe what I done. Was a frustrating journey. In short, I cut all pressing and hammered my back, especially 'middle back' like a mad man. I still do this, but have been in the clear for a while now (I now think it's how most should train - more pulling / posterior chain work).
Basically, when you think you have a problem with your shoulders, you should look at your shoulder blades.
Really, though, you could save yourself a lot of worry and try and get someone to look at it. Physiotherapist (physicaltherapist?). See if they can find anything wrong.
But in the meantime, I would advise you get away from the bar for a while, say 2-4 weeks. It won't kill you! Get away from the bar and OH pressing, decrease volume of pressing in general, do more rowing and see how you feel... I experiment with my own shoulder all the time which results occasionally in me pushing it too far. 4 weeks away from the bar and more rowing makes it feel 100%.
But I would rather hear that you went and got it looked at by a pro!