Neither a Dish Network nor DirecTV dish would, by itself, work for FM reception. If you connect the coax to you FM radio and receive any signal at all it is because of signal leaking in somewhere in the coax itself.
What you could do is use a diplexer at each end of the coax run, and mount an antenna for FM reception somewhere near your dish. The output of the antenna would connect to the "ANT" input of the diplexer, and the dish output would go to the "SAT" input of the diplexer. The connections are similar on the other end of the run inside your home, with the "ANT" output of the diplexer going to your FM radio, and the "SAT" output going to your satellite receiver.