My first thought is the footswitch jack itself. If you are NOT using a footswitch, there is a cutout contact in the jack that completes a circuit through the panel switches. WITH a footswitch, it cuts those switches out and ignores them. If that cutout contact gets iffy or the solder cracks under the jack, you can get similar symptoms.
If you ARE using the FS, the jack still can cause these problems.
Without the FS, check the panel channel switch with an ohm meter.
The Fender one wire FS system puts an AC voltage on the FS jack, then various diodes and zeners switch in and out and then op amps decode the result to control the amp.
Isolate the problem. Random channel changes could mean the control circuit is directing the signal circuits to do it, OR the control circuit is OK, but the signal switching is screwed up. The question to answer is this: when the channel switches itself, do the indicator lights on the panel shift or stay steady? If they shift, that means it is in the control circuits - like my FS jack notion - and the signal switching is just following orders from it. If the lights stay where they should be, then you have a signal switching element at fault.
And another question: is it really actually changing channels? Or is it more a matter of the channel stays put but the gain makes major changes?