If I factory reset will the games be saved onto the Xbox with a different account

Hey there xMikeTheRipper!

First thing, keep in mind that so long as you've been connected to Xbox Live your saves are tied to your Gamertag, so resetting to factory settings would only delete the game's install and not your saves. Have you tried downloading the offline update again
onto a different USB drive to see if that helps? Make sure you're also downloading the correct file, as there are two different Offline Updates to choose from. Keep us in the loop with what's going on :0

Try performing a
factory reset keeping your games and apps.

I will need to make sure the person I am sharing games with is aware of it so I can go onto their account and do the home xbox thing, other than that. Is there another method?

Games on the Xbox do save to the cloud on your account, so doing a partial factory reset will remove most things, but your saves will safely be stored in the cloud.

Saves are stored on the Cloud, the game data is erased when performing a factory reset. If you use that account on any Xbox, the saves are automatically downloaded from the cloud.

Factory default worked perfectly. I can finally re-establish my Artifact Assault dominance.

If you don't know how to do a Factory reset go to Settings > System > Restore Factory Defaults.

This does mean you will have to re-install all games onto your xbox which can take some time but all your data will be saved.