A similar prob kept happening to me last Saturday at HRH. At an idle, my car would start accelerating abruptly on its own. If I turned off the my TX and back on, it would work fine again. This happened quite a few times. I'm running a Spektrum SR3500 micro rx.
To resolve the issue, all I did was disconnect all of my servos and went through the whole initial binding procedure again. After I did that, the problem went away. One thing to note about 2.4GHZ wideband or frequency hopping narrowband radio systems is that the RF channels CAN and WILL saturate if there are enough radios turned on and can throw things out of wack. Being that Spektrum is the most popular 2.4 system, it's not impossible to saturate the channels that it uses during a big race. But anyway, going through the re-binding procedure always helps.
It really sounds like an RF interference issue though since it would work fine when you turn your TX on and off. If it was a power delivery issue, I would think that the issue would continue regardless of whether your TX is on or not. When your TX is off, the RX is still sending PWM signals on all channels on your servos. The only difference is that it's sending the PWM signal encoded for your failsafe positions.