First thing I want to know when this happens is what an external counter reads.
I always have a counter in line with the radio's output. If the frequency is almost the same as the channel frequency it shows in AM mode, that points to the radio's sideband modulator and audio circuits.
If the frequency is way out in left field, not anywhere near your channel frequency this means the radio's final and/or driver section is oscillating. When this happens, the frequency of the carrier is not controlled by the channel selector.
Loose ground screws around the edge of the main circuit board can do this.
And it could be that the power supply has a problem when the transmitter loads it down. If the power supply has what they like to call "hum", or AC ripple, this can get into the radio's mike-audio circuit.
Do you have another radio you can use to listen to this one and see what you hear when the Roadtalker transmits on sideband?
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