It sounds as though you have connected your tv to a composite connection on your second tv. Since you have a stable picture on that tv, I would say your tv is working correctly (you will have connected the green out to yellow in and green out carries the component sync signal, so the tv seems to be generating its sync signal properly)
HDMI is intelligent, your tv will tell the tv which resolutions it supports and only display the options it can use. Component is not intelligent and hence your tv will display all of the outputs it has regardless of whether they are compatible with your tv or not.
I suppose it is possible to select the wrong option and obtain the rolling you are experiencing, however I would ordinarily expect a blank or blue screen on your tv and after a few seconds the tv would reset itself to a compatible resolution.
So could it be that:
Either your second tv is behaving as expected and your tv is setting itself to a compatible resolution or that simply your second tv supports both PAL and NTSC signals. Additionally could it also be that your first tv, although not compatible with the wrong type of signal is not actually rejecting it and hence the tv is not trying to reset its resolution. If this is the case and you are unable to see properly to alter your resolution because of rolling, then you could attach it to your second tv and set it from there. Given the information you have given I'm guessing the resolution you need is 480i/60 Hz.
Failing this you are likely looking at your connections again, but at least you know your tv is not defective.