I am currently clearing out my memory cache in preparation for Skyrim. When I came to Fallout 3 however I really didn't want to lose
any of my DLC. So I was wondering if I were to buy the GOTY edition which, to my understanding, comes ready with all the DLC,
after deleting the content from my Xbox 360 would I still be able to play on my current save game if the content isn't on the hardrive
itself?

Yeah, you will still be able to play it. Just make sure you delete the DLC, and not your game save and you will be golden.

Why, you’ve paid for the DLC so you can download it as much as you want. Iv deleted DLC on my consol sometimes and all I have to do is hit the marketplace to get it back. (FREE)

I think the GOTY edition comes with a separate disk with all the DLCs on it, at least it did for the PC. I bought FO3 for the Xbox years ago and downloaded all my DLCs via Marketplace before they released a GOTY.

I think it would just be better to leave the DLCs on your HD in my opinion.

If you try to play with the add-ons removed then any equipment, perks, etc you got from the add-ons will be gone (you get a warning about this if you try it), and obviously you can't get to any of the add-on locations. However, all you have to do is reinstall the add-ons before you try to play again and all will be good. I've removed and re-installed all of them several times (xbox), as I've put the game down and picked it back up a few months later.

The GOTY edition comes with the add-ons on a second disc, and I believe that they do have to be installed in order to play them because the game will only play from the main disc. But as Arrow points out, any content you have purchased from the marketplace can be downloaded again at any time for free, so there is really no need to go out and buy the GOTY edition.