If there is any water in the garbage disposal sink when I start the disposal, pressure in the plumbing will blow trash and dirty water back through the other sink. The force is great enough to blow the stopper out of second sink, even with the sink is half full. I can kill the power on the dissposal for a couple seconds and back on and things get back to normal. Water will drain properly from the non-disposal sink.

You have a partial blockage in your drain line setting up a blow back situation. Let me explain. If the disposal doesn't drain right the disposal table will act like a centrifugal pump and since it's blocked from draining it pumps the discharge up into the other side of the sink.
The solution would be to remove the "J" bend from the trap and send a snake up into the wall. Put out 7 feet of snake. Flush with a pan of boiling water. Good luck, Tom