'91 300SE electrical problem

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Driving down the highway in rush hour and my car starts acting odd. Radio freezes and the car starts to hesitate. A few moments later the ABS and SRS lights come on, the lights are dimming, and the heater fan stops. I turn my blinker on to move to the right lane and the engine dies. I immediately turn the blinker off and it it runs a little longer, then completely loses power. There was no alternator light so I assume a dead battery (It's -6 degrees F here in Minnesota !). So, a few hours later I put a new battery in and it starts right up as expected. I drive most of the way home (about 20 minutes) and the same thing happens. I make it to a parking lot just as the headlights are dimming out of sight and the fuel pump is barely pumping. In total frustration I sit in the car waiting for a ride. About 20 minutes later (by now I'm freezing my buns off) I make an attempt to start the car knowing it's a waste of time. Surprisingly it starts right up. Lights are bright, etc. So, I drop it in gear and head home as fast as I can. As I pull into the driveway the process starts to repeat and as I stop in the garage the car dies. Any suggestions ? I am taking a shot at it and have ordered a new OVP relay.