How can I reliably play Fallout 3 DLC without an Internet connection?
I just moved to a new apartment, and I popped Fallout 3 into my 360 before the cable company got around to turning my Internet service on. Before the move, when I did have an Internet connection, I completed two DLCs (The Pitt and Broken Steel); I had some items and side quests left over from those (spoiler: e.g. the gun The Infiltrator and the side-quest about selling cameras to the BoS).
After the move, I opened up my old save file and wandered around the DC Wasteland some, but didn't do any quests. I then saved and turned off my 360. My Internet service was turned on the next day. After that, when I opened my save file, it looked like I had never completed either DLC; the unique items were not in my inventory and I was hearing Wehrner's Distress Signal again.
How can I get my items back? Or, if that's not possible, how can I play Fallout 3 while not connected to the Internet without losing my progress?
Assuming you didn't change Xbox profiles or try to move game files to another account, Xbox Live doesn't store saved game information and would not have caused this problem. (Your Xbox profile still has FO3 achievements granted for those DLCs correct?)
It sounds like you either overwrote your saved game, or you loaded an older save file. If you never reset the clock on your Xbox after your move (and it won't be correct with no Internet on its own), then it might be loading the wrong saved game file as it probably uses the timestamps for "Continuing" your game.
Try looking at all of your FO3 saved game files to try to find one with your progress (or at least closer to where you left off).
If the Xbox isn't the one you originally purchased the content on (and you hadn't migrated the licenses to the new Xbox), you would need to be on LIVE for the DLC to load (isn't DRM brilliant?).
However, if this was the case the game should have warned you as you were loading your save game.
If you can, going back to a previous save game should fix things.