Outside of arrivals, you can run from terminal to terminal. The PHL thread has the map stickied to the top, and it would be on above of the orange sections on the map -> AA Guide: PHL / Philadelphia International Airport - MCT, Connection, MCT, etc.

I don't remember seeing showers in the AC, and I'm doubtful it would have one considering it's a very old lounge; no clue on the BA side though but you should give them a call. Depending on the time of day, it's high 80's to low 90's, and it's very humid.