The '50 Plymouth, unless changed out, has ball and trunion U-joints fore and aft, also called Detroit U-joints. These can and do wear out. Most frequently the rubber sleeve covering the U-joint ages and cracks or tears. Then the grease inside can be lost or contaminated.
Inside the U-joints is a smaller shaft, with a ball on the end. The ball is bisected by a trunion pin. On the trunnion pin ends two more balls are fitted, which ride on needle bearings so they can rotate on the pin. The whole assemble then rides inside a heavy cast and machined housing, which has grooves to accept the needle bearing equipped trunion balls. So the U-joint can flex in the with regard to the driveline radial direction as well as move in and out in the driveline axial direction.
Proper rebuilding of the Detroit U-joint, including replacement of the rubber sleeve, involves pressing out the trunnion pin on a big hydraulic press. An automotive machine shop should have one, but you might have to look around for a machinist or mechanic willing to press out the pin and/or rebuild the U-joint.
When you get under the car you can inspect the rubber sleeves for integrity, and see if the driveshaft can be moved by hand forward and backward. If not, it might be seized up.
Usually the noise, I'm told, is a loud clunk when going over bumps and/or under load.
What I've seen is that the heavy machined casting that the trunnion runs in develops wear spots where the trunnion balls usually ride. Or there's a rubber sleeve/lubrication failure.
It's not too difficult to pull the driveshaft with the U-joints attached, but of course it's dirty uncomfortable work on your back under a car jacked up and on stands. (Don't rely only on jacks!). After that fixing the damn Detroit U-joints is no picnic, and pressing out the trunnion pin can be dangerous involving very high pressures. Pressing in a new pin requires a careful technique, because you want it to be centered on receiving ball. One mechanic who pressed mine in machined a spacer so that the pressing would stop at the right point.
Have fun.
