Select the files. Right click > Library Tools > Rename, Move, and Copy Files. Use the Find and Replace mode to replace E:\ with F:\ and apply the change. This will preserve the exact current directory and filename structure, but move them from one drive to the other.
If you prefer, you can also use this tool to re-organize the files into a new structure of Directories and Filenames based on their current metadata fields.
Check the Preview pane carefully before you apply the changes, and do a few and make sure you get what you want before you do them all.
Using a program, other than using the OS, makes me a bit nervous when moving files. IE, I'd rather copy, then verify, then delete the originals. However, using this method would tag all the files with new import dates, as well as "never played", ..., etc.
Many cataloging programs will allow for 'finding the file(s)' when the file(s) are missing. How can I move files "my way"??
Tia, and Happy Holidays!!