I will assume, by the description, that you're hooking this to a CRT (old tube TV). :p
The last time I had a console do what you describe, it turned out to be the AV input connection on the TV itself. While possible it is the cable, the symptoms usually ended up being different that you describe. Swapping cables, if you have a spare available, would be a great way to actually rule this out.
When you adjust the cable's position (aka "jiggling"), does it make a difference? Does it matter where along the cable you do this?
Check to see if the jack(s) is (are) loose in any way on the TV. If so, then the soldering that holds it may have cracked or broken in the TV. It might be fixable, but it's insanely dangerous and probably not worthwhile. (Met an electronics veteran who was lucky to be alive after accidentally brushing against the leads of a CRT capacitor early in his career. NOT RECOMMENDED!)
Your best bet, if the above mentioned diagnostic "jiggling" doesn't narrow down the cause, is to connect the Saturn to another TV, one that you know works. If you cannot reproduce the problem, then at least you know it's not the console or the cable. (Though honestly, I've never seen this problem be caused by a faulty or damaged console.)