Your CGI code looks ok in principle, and is definitely the way to do it - as recently explained at length elsewhere, LWP is a module for retrieving pages, not from creating them and dealing with their submissions.

This may be simply to do with exactly *where* you're running the script from and printing the result...if you're submitting from a browser, and want the script to print back to the browser, you'll need to stick a print $fdata->header() before that print statement, so that the browser will be able to display the result.
Cheers,
Ben