I need to update FWE from version 5.03 to 5.04, but I can't find any information on how to do that, and I don't want to risk messing up my files. Do I just download the latest version and get rid of the old .esm, or what? Also, my saves I made with the 5.03 version will be compatible with the newer version, right?

Yes old saves are compatible.
All you have to do is install the 5.04 update and overwrite any existing files.

(and of course readjust load order)

it's just a minor update, you can copy the contents over your installation.
but on a side note: use http://www.fallout3nexus.com/downloads/file.php?id=640! it makes it just so damn easy to revert any mess you've made.

I do use FOMM, and I love it.

Thanks for the replies.