If you have an infinite loading screen in New Vegas then I have a fix that should work without having to clear the cache!

(Before I begin I assume you have the game and all DLC installed to the HDD, this may not work if you do not have the game installed, although I am unsure if DLC has any effect on the methods below)

First you'll need a flash drive(or USB storage unit), it doesn't matter what size just as long as it can store you're save games. Then before you play the game, you need to go to Settings > Console Settings > Storage and go to the drive that as your New Vegas save file(s). Go to a save file, press A then go to Copy and copy it to a flash drive. Make sure you do this with all saves, auto saves etc. (You won't need to do this with DLC if it's installed on the HDD) MAKE SURE YOU "COPY" IT NOT MOVE IT!

When you start the game, select the drive which has your DLC and game installed on (This is usually the HDD). Then go to "Load" and press X, and then choose the flash drive/storage unit which you copied your save's too. Then load up your game as normal.

The loading screen will be a little bit longer than usual so give it about 60 seconds, and if all has gone right it should load the game correctly.
This can be done without clearing the cache, so you shouldn't need to worry about downloaing massive patches for your games.

You can try saving to the HDD again if you so desire, which I recommend in case you need to do this again in the future.

Also before saving make sure you holster your weapon as this seems to cause the issue in the first place.

Its also good idea to use more than one save, use at least two saves, one auto save, and one manual save, however do not rely on autosave too much. Instead only use Autosave as a back up in case you forget to manual save.

Note: If you have the 4GB model Xbox and have an HDD installed, then copy the saves over to the 4GB memory unit instead of a flash drive. (This is what I did) You may also try deleteing any stuff you don't want from the HDD, which should help.

However if you do not have a HDD installed then you should use a flash drive or external HDD.

PS3 Note: The same method MIGHT work. Just do the same, copy any New Vegas Save games to a flash drive. Although I cannot be 100% sure on this since the PS3 version has it's own issues. I also don't know if you can load up save games from a flash drive on the PS3 version, since the save games may be copy protected.

I hoped this has helped. I've tired and tested this a few seconds ago and it worked like a charm. As I stated the loading screen may be longer than usual, so don't panic and give it some time.