Oh! A couple other tools that might be useful to you, depending.
The WordPress option: like I said, you can just set up a WordPress account, use the built-in Tumblr import tool, and point it at your Tumblr blog to copy all the Tumblr posts over to a WordPress blog.
Frostbox: this one comes with a subscriber fee, but it lets you back up all sorts of things, like various social media accounts (I don’t know if you can move them between accounts): http://www.frostbox.com/this-is-frostbox
Backup Jammy: This one just snags all your Tumblr posts and turns them into an HTML file that you can save, so you can just keep them on your hard drive. http://boutofcontext.com/tumblr_backup.php
A few tools that let you back up various files from your account to save them on your hard drive: http://tumblring.net/how-to-backup-your-tumblr-blog/
The thing is that most of the backup tools just link to videos and images and stuff. So if you actually want to keep them (in the event of, say, Tumblr deleting something), you have to back them up some other way.
BUT this also turns out to explain what’s up with all those art reposter accounts. Because what the image backup apps do is, you just point them at a blog and then it scrapes all the images and saves them in a folder on your computer. So then they’re just…there. And if you’re the type of person who does this, you could simply dump them back onto your art-reposting blog.