If this is the case, Bethesda you should seriously hire me as a bug tester. I don't even have to find the bugs, they find me. They find me and ruin me.

Let's start from the stop shall we?
I'm not entirely sure if this only has to do with Dragonborn, but the game had been acting kind of funny the entire time while I was playing it, and I had been having issues with it freezing more than normal. I just finished the main quest line in the DLC and received the quest from the guy doing research on the Skaal to explore the cave he found. The game froze shortly after, but I figure whatever, it was a good place to stop anyways, so I turned off the game... never to return again.
So the next time I turn on my xbox, things are moving a little slower than usual, everything from the loading of my gamertag to the loading add-ons... the loading add-ons to the point where the screen froze, forcing me to turn it off. Turing things off while loading is never a good thing, which I realize, but when your hand is forced, it's well... forced. And it's been down the hill from there. The menu froze whenever I tried to load the add-ons. I deleted Dragonborn, still didn't work. Deleted Hearthfire, didn't work. Deleted Dawngaurd....didn't work. So no DLC on my hard drive at this point. Reload dlc, problem still not solved. Delete dlc again, try and load game, still freezing but this time at the start screen. Yep. Can't even press start.
So then I start deleting saves... slowly watching my life in skyrim slip away into oblivion (pun partially intended). I leave one hopeful save, go to start the game again and.... Black screen. We don't even make it to the start screen. HAH!
I pop in Oblivion to see what would happen... and I get errors saying key features are missing from my xbox. Right.
Well, I'm still working on the issue, and it doesn't seem like anyone else has had this problem. Go figure.

I love Dragonborn. I love Skyrim. Hate that this happened. Have been reduced to manic laughter at the situation. But I'm crossing fingers that either I figure out a way, or someone else does, to save my game memory.

try a different game
i mean really different, like halo or a racing game or something
if its acting the same, its probably just your xbox on its last legs getting ready to die
those things you described are all common occurrences on a 360 getting ready to go red ring
doubt a dlc would be 100% to blame for corrupting a hd
just doesnt work like that with the hardware/software of consoles
dragborn may have hit a final kink in terms of working the 360 to its breaking point, but i really doubt it is entirely to blame, as in doing virus-like damage
how old is your xbox? not that matters completely since ive seen 360s die after just weeks of use

It is probably just your HDD dying that caused it. I too suffered this last year, my HDD died taking 6 years of data to silicon heaven.

Consoles don't last forever~ If it acts the same when playing another game, or the console is slow entirely, then your console is on its death bed~

Which model do you have? Is it one of the model that you can buy a new drive and transfer the data from old drive? Microsoft do sell a data transfer cable. My daughter's Xbox360 Elite died last year and I was able to transfer everything from the old drive to a new Xbox360 Slim. You may want to do this before the drive actually fail.

do you use the cloud save feature? if not u can try that option.

The menu freezing when you load add-ons is normal, it can take a while, it does not mean anything is wrong, naturally it is going to take longer with more DLC. Turning the console off while loading could have caused problems. The game freezing a lot is just a symptom of an overworked game engine, you can cure most of it for a while if you want to clean the system cache and do a fresh install of all your Skyrim content. You will still get the occasional freeze on loading screens. Does not mean there is anything wrong with your hard drive.

Are you actually connected to the internet to download patches? I doubt there is anything missing from your XBox, Oblivion was most likely complaining it was missing DLC, which you had previously deleted. You could have just moved your saves to another storage device rather than just deleting them. Your saves would have no affect on anything unless you actually loaded one in the game. you would have had plenty of free space deleting everything.

Clean the system cache, delete Skyrim if it is installed, the Skyrim patch and any DLC. Probably restart the console to be sure, install Skyrim (making sure your disc is clean and readable) install the update and then redownload all the DLC and start a fresh game and see what happens. And just wait while the DLC loads, a frozen screen is nothing to worry about unless it goes for something like 15 minutes. It usually takes no where near that long to load.