If the 13 MHz frequency you're seeing is exactly half the channel frequency, it's the radio's PLL design. I don't have that radio's diagram handy, but a common way to simplify the PLL for AM-only CBs is to run it at half the channel frequency. This is meant to avoid problems with the final stage leaking RF energy back into the PLL. If it runs at your channel frequency, this causes "pulling", or a tendency for the frequency to wobble around with modulation. A PLL that runs at half the channel frequency is less susceptible to this quirk. A frequency doubler circuit should convert all the 13.5 MHz energy to 27 MHz. More to the point, should also filter it from escaping from inside the radio.
If someone was tweakerizing internal adjustments, maybe it could cause this?
73