1995 Chevy Tahoe ABS & Brake Light Comes on but Power F

My 1995 Chevy Tahoe just started something fun. The ABS and Brake indicator lights have started to slowly flash and then there's a loss of electrical power every minute or so when driving (enough to reset the radio clock). When I come to a complete stop, whether in drive or park, the lights go off and no power fluctuation. However, if I turn off the car when this is going on, the battery is dead and I have to get it jumpstarted.

My question is where to start looking? Is this a Brake/ABS short, an alternator or a battery issue? What would you suggest looking in to?

Thanks!