If in fact your battery is at full charge, meaning that it will show a minimum of 13.8 volts, and maintain greater than 12 volts under a load test (not just showing 12 volts at "rest" with no load), then it's most likely either a ground issue or a/more than one, loose connection in the starter circuit. The ground issue frequently shows up on new or rebuilt engines because the parts are freshly painted and the starter housing doesn't get a good ground circuit from it's housing, through the engine block, and on to the frame. Also, you should have a ground strap from one of the starter bolts, or at least one of the bellhousing bolts to the frame. Also you mentioned a solenoid on the "firewall". That is more common to Ford. The Chev solenoid is the smaller cylindrical shaped item on the top of the starter motor. If when you changed starters you transferred that from the "old" one to the "new" one, it could be the solenoid that's defective. But I'd start from the top of this list and work down to that rather than jumping to conclusions.