Then you should be able to disconnect those four lines, one at a time, and the noise will still be present. And if you remove the mosfet, everything should turn off. All of that would tend to confirm that the mosfet is the problem.
But if the mosfet is fully off when you power up initially, and turns fully on with no problem, it is highly unlikely there is anything wrong with it. It seems much more likely the .7V is coming from the radio, flowing backwards though the 3.3V regulator to the drain. Anyway, if you can, you need to just try removing or disconnecting things until you figure it out.
It would be helpful if you could measure the voltage at some of the important points, and report what they are when things are still On but should be Off. Like - those four I/O pins, and the Vcc pin of the radio module.