Okay, I've got one more suggestion before we turn to the video switch or whatever it's called. Does your TV have an "AUX 3/4 IN" and, if so, are you screwing your RF switch into it? If you are, you need to switch your TV to its "AUX 3/4 IN" mode or whatever it may be called on your TV. Switching it to its "AUX 3/4 IN" mode should be just like switching it into the composite video in setting except it's not composite.
Ideally, you should remove the Cable TV or Antenna cable from the back of the TV, screw it into the Nintendo RF Switchbox, and then screw the switchbox's cable into the TV. When you turn your system on, it should override the Cable TV or Antenna signal and display the game's signal (that is, if everything's on the right channel). That way you won't need to worry about the "AUX 3/4 IN" and have the problem you're having. I never use the "AUX 3/4 IN" so I can't tell you exactly how much of a problem it can cause.
Since you can get good video and sound from composite cables out of the A/V multi-out, you should be able to get good RF from it when you use the modulator that plugs into the multi-out. I've never tried it on an SNES, but I see no reason why it shouldn't work. I doubt that you have a bad modulator, too. You've just got to make sure everything's set to the correct channel. Even if the RF out in the system itself is fried, the external modulator independently mimics the hardware that is in the internal RF out inside the system itself, so there should be no problem using the external modulator because it acts on its own.
When in doubt, set everything that goes to your TV to channel 3 (or the appropriate channel in your area). That way you can test each device separately.