Hi cvlewis, welcome to the forums
Hmm, really not sure what could be causing your problems here, I used to hook up my Gamecube with component cable but never had any problems with it.
Assuming it must be the official Nintendo component cable as I'm sure there were never any 3rd party component cables produced for the Gamecube.
I really can't remember if you need to change any console settings when first changing over to component, don't have my Gamecube connected at the moment to check. I remember having to adjust console settings on the PS2 when changing over to component though.