Hi Spencer,
It depends on the level you have with Brinkster. With Premium levels you can FTP upload - and all links will change as long as you have the site defined within UD. But if you have the General membership (free one) you have to use Brinkster's file upload tool. And you will have to change the links to reflect. If this is the case, you can do a edit and replace for your entire website. For example search for something like c:\website\images\picture.gif and replace with ../images/
But be careful, you want to be as specific as you can to identify the text to replace - otherwise you run the risk of changing a whole ton of links - to only need to change them back to what they really are.
If you have the General membership, I can point you into a direction of a very reasonable host for hobby or personal stuff, giving you the flexibility that paying hosts do (at a huge fraction of the cost). - no I am not affiliated with them at all, I found them - and have a hobby site there as well.