It is not the bindings.
Try and keep both styles going if you can. If you can switch Goofy to Natural and feel comfortable doing both it will make life a lot easier when you start different tricks and doing fakies.
Having the board drift/rotate under you could be caused by a few different things. Firstly , it is hard for the board to get out of track and rotate if you are using an edge. If you are flat boarding then there is nothing to hold the board from rotating and it WILL rotate. So try and keep an edge (toeside/heelside), it doesn;t have to be a massive one, just one that keeps you stable. Thats why you see a lot of people doing small "S" moves as they go along, this is keepoing an edge. If you have an edge you have control, control to turn and control to stop.
I caught a massive edge trying to get up some serious speed to get through a long flat bit between two runs last month. I intentionally flat boarded to decrease my drag but ended up rotatng and catching a toeside edge. Thank god for my helmet. Though it still took me almost 2 mins to collect my thoughts and catch my breath afterwards. The helmet cam video is hilarious....
Secondly, you will stand more naturally when you are correctly footed versus fakie which ever that turns out to be(goofy or natural). Meaning your shoulder will be more side on and your head rotated correctly to the front. If you are actually doing a fakie and not realising it as you don't know which is your correct side yet, then there is a chance your body will be rotating a little more forward to compensate. This will cause the board to want to rotate too and there you have it again.
Remember the board will rotate based on your shoulders. Turn your shoulder and the board will follow. So if you are actually riding fakie then there is a chance your shoulders are rotaing slightly forward.
Hope this helps. Good Luck.