Check wires and connections to the starter itself. Sounds like you aren't getting a ground. But that would only solve the starting problem, not the cranking problem... No crank, no power, and the RCM can be a problem. Not starting right away while cranking, is a fuel problem usually. Whether fuel pump, regulator, throttle body, etc... You do have more than one problem now...but, leave the new battery in...the old one is on it's last days...good luck