It doesn't sound like it's damaged.... are you certain that your TV is expecting an NTSC signal, though?
Try booting your A1200 with both mouse buttons held down. Then go to Graphic Options. It should have "NTSC" selected currently (if yours is indeed an NTSC machine), so try changing it to PAL and then launch the game - maybe it'll act differently.
It's also possible the signal is degraded because the capacitors are too old (very common in A600, A1200s and A4000s particularly) as this could prevent the sync signal coming through cleanly.
Incidentally, a monitor can't use RF - you have it the wrong way round - your TV can use RF, a monitor can't. Did you try RF on your Amiga with the TV? If it's channel 3 or 4, it's American - if it's channel 36 it's UK.
Your best bet, though, is to get an RGB->SCART cable if your TV has SCART. I'd guess if you're in the USA, though, you may well not have a SCART socket as for some reason they never caught on over there (which is daft because these days all the TVs have the same electronics inside - there's just no connector probably - and SCART is really useful!)
Hope that helps a little - feel free to ask more questions (and let us know how you get on!)
