How to turn analog audio into analog cable signal for radio receiver?
OK now I'm confident to answer this: -

Analogue cable radio doesn't use FM or AM transmission - it uses data from a coax coming into your home.

If you do have analogue radios that work the traditional way you can buy small FM transmitters (intended for iPhone to car radio connections) just like these

You can get cheaper ones that just take an audi input jack too.

If you are still intent on making an FM transmitter (mono for simplicity) here's a link to a design site with several on.