If it is a 1995 EZGO cart, the controller has F1&F2 terminals and a 10-pin connector on top, and a motor without a speed sensor, I'm pretty sure it is a DCS drive system rather than a PDS.
Since it was driven home on bad batteries and a DCS controller has no low voltage cutoff, it is very possible the controller was all but destroyed on the trip home.
Since the solenoid clicks and the third wire coming out of the backside of the charge receptacle is hanging loose, the reed switch has been bypassed somehow. Also, the solenoid clicking indicates the controller is powering up.
It was mentioned that full voltage from the new batteries is only 37V, so I suspect they have not received a full charge yet. While 37V is not enough to drive cart very far, it is enough to troubleshoot.
Connect your DVM between the B- and M- terminals on the controller.
Press pedal just far enough for the solenoid to click and you should read full battery pack voltage (apparently only 37V now, should be 38.2V when batteries are fully charged.)
If you get full pack voltage, slowly press pedal to floor and the voltage between B- and M- should smoothly decrease to Zero Volts.
If it doesn't either the controller, or the throttle input to the controller is bad.
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If your F/R switch has a rubber boot over it (see picture), the attached schematic is the one for your cart.