Firstly, congratulations to the thread starter for managing to swim at all. I had a similar background and up until a year ago choked terribly even holding my breath stationary under the water! My TI coach gave me a series of exercises to work on the problem, and after a month I was able to stay submerged holding my breath without choking, then after another week of practise I could swim under water holding my breath. This then gave me confidence to swim underwater normally and control my breathing. We think I was probably inhaling through my nose subconciously when under the water. I'm absolutely fine now and have no issue with water inhalation, and I even manage to breath with one eye below the surface and one above, to maintain my streamlining.
In answer to the question though, I agree with everyone else, it's a matter of timing. Bear in mind that even when your nose is clear of the water when you turn to take a breath, they'll be a tangible split second where you still need to breath out to clear the water that gets dragged up by your face due to surface tension. This is something I'm conciously aware of. If you breath properly though there's plenty of time before the next pull and rotation to get air.
In answer to the moving/stationary question, this is perfectly normal and you will have to try harder to stop water going up your nose when you're moving.
Hope that helps.