Battery keeps going dead

The generator needs to be professionally load-tested on the car. In-store test benches aren't strong enough to do much more than what you did with the voltmeter. All they're going to show is the unit is developing some current but it won't tell you how much.

It is fairly common for one of the six internal diodes to fail. When one does the generator will only be able to develop exactly one third of its rated maximum current. For the common 90 amp generator that means 30 amps which is not enough to run the entire electrical system under all conditions. "Ripple" voltage will also be very high. The battery will smooth out the ripple voltage so you can't measure that directly with a digital voltmeter. That is actually calculated by the load tester as it reads the pulsing current coming out and going to the battery.

You need to have it load-tested on the car to see how much current it can produce. The voltage can be fine but if it can't produce enough current the battery will have to make up the difference until it runs down.