park to reverse, the transmission feels like to gets picked up and drops into place

It sounds like you have a bad transmission mount. The exhaust and the cross member will need to be removed to replace the mount. Here is a picture of the mount looking from under the vehicle.

So changing that will fix the issue of the car feeling like it’s being lifted and sat to place while changing gears? You can notice the rear moving.

Yes, that is the job of the transmission mount, to keep the transmission in place during these situations. The engine mounts should be inspected as well.

I’ve replaced the transmission mount but the movement is still there. It seems like the U joints need a replacing.

Is it the transmission moving or does it feel like the rear of the car is actually moving? When you replaced the transmission mount what did the cross member it mounts onto look like, solid and no rust and the chassis looks the same? Is there any clunking, squeaking or other noises when the car moves to indicate a failed U-Joint? Think you could take a video of the engine as it is shifted between those gears, it is possible it's a failed engine mount if it is the transmission moving a lot.

The rear moves a bit when changing gears. It makes a clunky noise and It squeaks when driving. The only movement I feel is on the rear but I also hear squeaking under my seat as well.

Okay, in that case I would check the U-Joints, it sounds like at least one is binding very hard. I generally like to replace both of them if I have the pull the driveshaft, they both have seen the same amount of wear and tear, and it's easy to do while you have the shaft out anyway.

So I’m underneath the car, one side is connected to the differential with bolts and the other end is straight to the transmission without no bolts. I can twist the driveshaft and there’s play in the part where it goes directly into the transmission.

The part that goes into the transmission is the front slip yoke. It should be able to slip in and out but you shouldn't be able to move it up and down or side to side more than a tiny amount.
https://www.2carpros.com/articles/how-to-remove-a-drive-shaft
https://www.2carpros.com/articles/how-u-joints-work

I’ve checked both U joints and they look pretty good still. There’s no play from the 2 joints now. I’ve greased both joints and the movement is still there. I am now confused on what the problem can be.

I will try removing the U joints tomorrow and replacing them with brand new ones but I doubt that would make a difference.

No play doesn't mean one isn't damaged and causing problems. I've seen them look great from the outside and still be bad. You should be able to feel it though once you pull the rear yoke free. Try to move the joint in both directions, the bearing caps should move smooth and free.

One thing I did notice was when I was using force to push the “front slip yoke” upwards, it moved the transmission. Maybe that can also be a cause? I’ve just replaced the transmission mount.

About the u joints, I will swap them out and let you know the results.

Have you had any success or found a problem? Or just not had any time, I know how that is.

Unfortunately no. I haven’t had any time.

No problem. These days the time seems to get used up fast.

So it seems like the passenger side upper control arm frame is completely broken. I had it welded on and it helped somewhat. I am now going to install toque box reinforcements and see the results. Also when I have the car lifted and I spin the rear wheels from left to right. I hear a clunk where the transmission and driveshaft meet. Is that normal?

A slight clunk is usually play in the splines. I don't like to see any because it means the splines are very worn. The clunk is from the slack from the wear letting things move.

The A Arm broken free would make things interesting.

So can it be repaired and how? Is it expensive?

It can be repaired, how expensive depends on where the wear really is. If it is the splines in the front yoke then it's a simple thing to get a new yoke and change out the U-joint and yoke. I suspect it's a combination of wear in the yoke and the output shaft in the transmission. That will cost more but the only way to know is to check the various parts for wear. There needs to be some clearance in all of the parts or they will bind up but not an excess amount. To check it you would pull the drive shaft after checking the U joints for play, then you take the front yoke off and examine it and the output shaft for wear. The problem there is that without a known good part or a tester for them it isn't easy to tell if they are OK or not. I would see if you can find a transmission shop and have them look it over, they can tell you if it's excessive wear and usually have ways to test the parts.