Increasing blood in the area: Might try using a heating pad or mobile patch. Also a heating lotion/cream/rub. Wear a heavier shirt or light wrap when working out. Warm up the shoulder girdle well. A good shoulder stretch is just hanging from a overhead bar, allowing the body's weight to pull you down, in a relaxed position. Can also do this one arm at a time. Wonder if you have done different grip positions on the press?
If overhead pressing bothers you, than don't do them. Do the lateral raises standing , sitting, on an incline bench, flat bench, cables, etc, only if no pain is present. If you can do front raises, than do them. Also Hi-pulls (Gironda style) which is an excellent shoulder movement. Might also check you style in the BP. Sometimes this can affect the overhead pressing muscles. Good Luck.