You have multiple problems, I think.
Biggest question - when did this problem start vs repairs that were done?
My 1964 dodge custom 880 steering wheel and car (feels like back tires) shake when I take off.
I would presume this to be driveline related, possibly even excessively-worn rubber mounts (or they are permitting the shake, at least).
Once I go into next gear and over 30 mph, the steering wheel and back body wobble/shake stops.
I'm thinking driveline still - you are putting less torque thru it when going to higher trans gears.
When the new U-joint was installed - done by you or a pro?
When installing a new U-joint, pressing the cups into the driveshaft, it is possible to install it OK with the C-clips but with the driveshaft ears in a little bit of a bind, causing extra friction in the needle bearings.
Hitting one or both of the ears inward/outward with a hammer/punch can release that and make a noticeable difference in how the joint pivots. This doesn't happen with the 2 cups
When pulling away from a stop, the pinion gear is trying to climb the ring gear, and the nose of the axle is moving upward, leading to increased driveline angle at the rear U-joint.
Perhaps I'm grasping at straws, but if you have a tight-spot U-joint, at the rear, with the change in angle, and the 180° accel/decel nature of a U-joint - that could give a starting-out vibration.
Which would be lessened when you have less driveling torque and less axle windup when getting into 2nd gear.
I have a car with the B&T front joint but don't know a whole lot about them.
I believe you can disassemble it somewhat, clean and re-grease, and put some type of split-boot for a CV-joint on it.
Doing so, you might find excessive wear if it's gotten a lot of dirt inside?
But I've generally not heard of folks discussing problems with the B&T.
When I am cruising on the highway, the car drives very smooth. However, when I begin to stop, the car vibrates really bad. The car's front suspension has been done (tie rods, ball joints, bushings, etc). However, I have the original ball and trunion drive shaft and the boot is missing but I have a new u joint. I dont know the condition of my motor or transmission mounts. Lastly, I have drum brakes with one line all around. Right side rear back drum brakes gets really hot than the other drums and I can't get them off?? Should i consider a driveshaft conversion or can bad rear drum brakes cause this problem?
You have a brake problem there, at first-guess it's not related to the other problems.
Others have mentioned loose lugnuts (I've experienced a really bad shake from that), bad belts in tires (definitely causes a bad shake).
I would agree with those if it weren't that the car smooths out at highway speed.
I have a hard time believing that a front-end problem would cause a shake at low speeds but be 'very smooth' at highway speeds. If someone can explain that, I'm all ears.