Jeri99,
If this was a video of your truck, I also noted play in the rear u joint of your front drive shaft but that isn't your primary issue.
You never filmed the front u joint of the front shaft. Is there a reason for this?
Start with checking and replacing all u joints! Go with the cheap fix FIRST.
Your differential carrier seems quite greasy. Is there a leak somewhere on this carrier? Did you / do you have to add gear lube? Have you checked the level regularly / lately?
Is this transmission fluid drifting back from your tail stock seal leak? There was quite a bit of evidence on your tail stock and floor boards, is this u joint grease or transmission fluid?
At any rate, that noise i.m.h.o. is emanating from the rear differential carrier.
FIRST, set the parking brake and chock the wheels. Disconnect the battery. Put the transmission in neutral.
Next, check the gear lube level in the rear differential carrier. If it is low or not readable, this could be pointing to your problem. Replace the plug now so you don't forget.
Grab the drive shaft just in front of the differential yoke. Work the yoke clock wise and counter clock wise and note the range of play. A small amount is required / normal but if it is excessive, you could have a worn ring and pinion gear set. (You won't be able to detect bearing play at this stage.)
Also work the yoke side to side, up and down, in and out looking for movement. There should be very little if any. If you find more than just a minimal amount of play, this would point to / result in a worn ring and pinion.
If you cannot detect any movement in the yoke, your next step is to remove the rear joint from the differential carrier yoke being careful to hold the caps on the rear joint as you remove it so you don't disrupt the rollers inside the cap. Now tape the caps in place with electrical tape. Move the drive shaft aside, no need to take it out of the tail stock at this point.
Repeat the yoke movements above.
If movement is still not found your next step is to remove the differential cover first getting a pan to catch the gear lube. Once the cover is removed, look for discolored gears, chips in the bottom of the carrier and a step in the teeth of the ring gear. That is where the 'excessive wear' is happening IF there actually is any occurring and damage hasn't yet occurred. There might be a slight metallic sheen to the lube which in and of itself isn't out of line but if there is more than just a little metallic sheen, you might have more than ordinary wear going on.
In all of the above concerning the rear differential, if you consume gear lube via a leak and you or the previous owner (if any) haven't kept it topped of regularly this might be the cause of your damage.
If no damage is found wipe out the inside bottom of the carrier with a rag. Clean the rear cover. Replace the gasket with a new one if a gasket was used or 'form-a-gasket' per instructions. Refill with NEW, DON'T use the old gear lube!
If you do detect excessive wear or damage to the ring and pinion or elsewhere, skip the gear lube replacement. A yours is a 2002, you might be able to find a good replacement in a recycling yard for about $300 - $500. Much less than a new ring and pinion set installed and unless you or someone you know are well versed in auto mechanics, do NOT attempt a ring and pinion replacement, it won't go well and you'll soon end up once more where you are.
As for your hard shifting, that is purely a transmission issue of some sort. The clunk was, again i.m.h.o., in the rear differential, merely taking up the slack when it was finally reloaded after the hard shift in which case you might have two issues.
If you find that you in fact do have multiple issues, again being a 2002, you have a decision to make... is this rig worth keeping with two possible large and expensive issues to fix?
If so, your best route might be to find a good parts vehicle from which to take the 4160e and rear differential, just make sure you get THE SAME gear ratio as you have now. You CANNOT have two different ratios! If your parts vehicle has different axle ratios than your '02, you can also replace BOTH differentials to keep the ratios alike.
In closing, I had a '99 Tahoe that had a rear 'clunk' such as your '02 has. I checked it a short time before and found it had moderate play in the differential as well as some play in the yoke. I knew the differential was gone but decided to drive it until it let go... this was A BAD DECISION on my part! When the differential finally let go (3k miles later) it did so with a loud BANG! and very loud grinding / growling. The carrier cracked and gear lube went all over the underside of my truck and onto the highway.
When I opened the cover, gear lube, gear teeth and bearing rollers spilled out nto the pan. My differential was a locker.
The axle, towing and labor was over $1000. I sold the truck soon after replacement and it is still going strong.
Good luck.