Are you saying replacing the dead battery with a new one does not allow the engine to turn over? This sounds like a batter cable or starter circuit issue.
You can disconnect the battery wire from the starter and see what happens overnight. Better yet, pick up a low cost amp meter capable of handling 5-10amps and temporarily hook up it up between the positive battery cable and battery. Play around with the fuses, wiring etc. until the reading drops to minimum. There will also be a small drain for the radio memory, clock, and ECU memory. Don’t attempt to start the car or turn on any high amperage devices or the meter will be damaged.
Yes, the solenoid can be replaced separately and is not a difficult job. Check out
www.rockauto.com. Or call a few parts stores.