Your website says Camtasia Studio should be able to import/edit .mov files. The only type of file my Kodak video camera produces is .mov but Camtasia won't recognize those files when I try to import them and I get a "No available codec" error.
@Scotty: Before trying a converter, you could try changing the file extension from .F4V to .MP4. I don't have access to a file from Adobe Flash Media Live Encoder to try it for myself. But I'm pretty sure the .F4V file extension was a choice by Adobe to indicate an MP4 file that was compatible with the Flash Player.
Camtasia Studio supports MP4 files that use the MPEG-4 Part 12 container, and contain H.264 video, and AAC audio. So if your F4V file matches those (it might), then it should work if you change the file extension.