Maybe you didn't build it with a tight layout on a pcb. At such a high frequency as 100MHz FM, any piece of its wiring longer than a few mm adds capacitance to detune its frequency. I built my FM transmitter on Veroboard (stripboard) and it works fine. I used a very tight layout of its parts.
Many things change the tuned frequency of a 100MHz coil:
Its wire size, total length, length of its wires, inside diameter, outside diameter, number of turns, the spacing of the turns and the capacitance in parallel with it.
A microphone or music source on an FM transmitter doesn't cause it to make a new frequency. The audio just causes its RF frequency to swing above and below its tuned frequency a little. The swing is at the rate of the audio and is called Frequency Modulation (FM). Very loud audio causes a 100Mhz frequency to swing from 99.925MHz to 100.075MHz. With no modulation the frequency is 100MHz. With the microphone disconnected the frequency will remain at 100Mhz.