You never, not in AM and not in FM, listen at only one frequency. That would be a perfect filter. No, we always listen at a little piece of a band, this is called the bandwidth. So for example: you say you listen at 7MHz, but actually you're listening at 6.999 - 7.001 MHz.
Since the frequency doesn't change in AM, we can have a very narrow bandwidth: usually less than 250Hz, when we transmit CW, since that only has one tone. When we transmit speech over AM, the bandwidth has to be more, conventionally 9kHz in Europe.
In FM, we do change the frequency. So while the base wave is, let's say, 100MHz, this may vary from e.g. 99.05MHz to 100.05MHz. The bandwidth of phone is approximately 3kHz, but to allow higher frequencies as well the common bandwidth of FM is 15 - 20kHz. So when you say you listen at 100MHz, this would be from 99.99 to 100.01 MHz, with a bandwidth of 20kHz.
So for AM, we have a rather narrow band filter, while we have a rather broad filter for FM. This is, as you said, because FM changes the frequency, while AM changes the amplitudo.
More information: http://www.cybercollege.com/frtv/frtv017.htm (from "Basic Differences Between AM and FM" to the end)