Severe Drivetrain Shaking

My guess is the harmonic balancer slipped.

I take it that it does this when parked correct?
My guess is also the balancer, be advised if it is do not run the engine, period.

Double check the firing order, Make sure what CAM you have ( two different firing orders for a 302)..
PUll all plugs and take a compression test..
As said above check you Harmonic balancer......
Is this an external balance engine??? 28 oz or 50 oz???? See what weight ( or NO weight) is on your flywheel and balancer... Make sure you know what you need before you order any new parts....

You stated that "Whole drivetrain started shaking out of nowhere. Just got the car, so I’m not sure of its history." By this I am taking it that you drove the car for a period of time (if so how long was this?) and then the shake started and has since not gone away. If so then I would assume that the rotating assembly is all matched up at a zero, 28 oz or 50 oz imbalance so nothing to worry about there.

This would literally mean something has either come loose or slipped. If the flywheel or clutch set fasteners have backed off that is a problem but it sounds like you have driven it / ran the motor enough that if that was the case all hell would have broken loose. With that being said I am still on the harmonic balancer and it is an easy check that takes little effort. Bring the #1 cylinder up on TDC and see where the timing pointer is showing on the balancer. You may also rotate the motor over by hand (may have to remove the spark plugs to make life easier) and visually see the external ring on the balancer not lined up with the internal ring or the external ring just slipping on the internal ring if you have the serpentine belt on when you do this.