A few other things to check before replacing the solenoid.
Check the voltage on the small terminal(s). If one small terminal check for 12v to ground when the ignition switch is on, if 2 check for voltage between them with the ignition on.
If that's good, check that the voltage to ground on both large terminals is the same with the engine running. If the same, the solenoid is doing its job. If different (higher on the engine battery side) replace the solenoid.
Mine has a circuit breaker in the wire to the engine battery. That could also be the problem.