I will be losing my current hosting, and I need to move my Omeka site to a new location. I haven't seen any posts that specifically address this - any advice? I've been looking at the documentation about backing up and restoring your database, and moving items in your Omeka folder when upgrading - should those basic procedures work for me?
There are two main steps: back up your entire database, and back up ALL files, including the Omeka software itself (that means the archive directory, as well as application and all others). You should have no problems.
If you want to upgrade to the newest version of Omeka on your new server, I'd strongly suggest migrating all of the old data and getting your site up and running first. That will be the only way to know if there's a potential problem in migrating, or the upgrade process.