Ok. Two ways to do this.
Close down Eudora and edit the Eudora.ini file found in your Eudora folder. Look for a setting named TabooHeaders.
This is a list of headers which you do not want shown, sent or otherwise. Add to this X-Mailer and X-Sender, but NOTE that this entry has a limit of 250 characters MAX, so you may have to remove one of the sillier headers (like X-Juno) to get under the 250 limit. Save your changes, start up Eudora.
The second way is to open a new email and type in <x-eudora-option:TabooHeaders>, then hold down the ALT key and click on it. You can then edit the entries in the tiny box. Click to save.
With either method, your NEWLY sent messages will not have either either.