Hi Frances: the best thing you can do for right now is unclip at least 1 side if you even Think you might want to stop. That way you don't have too many things going on at once: stopping, unclipping, watching traffic, etc etc.
Also, if you leave 1 foot clipped in when you stop then you can use that foot to get you going again while you clip in the other side.
There has been quite a bit of discussion on which way to turn your front tire when you unclip, I would suggest trying both to see which works for you.............when I stop I turn my front tire in the direction of the unclipped foot, since I've started doing that I don't fall. Someone else suggested turning the front tire away from the unclipped foot, said it works perfect for her. You might pay attention next time you fall which direction your tire was turned and trying turning it the other way, see if that helps.
As for unclipping/starting on a grade: ride downhill a short way until you get both feet clipped in, then turn and go back to the climb.
The other thing that might help you a lot - practice unclipping/clipping without worrying about stopping......just to get your brain and your feet used to the whole process!
Just don't give up - we've All been there and some of us have ridden clipless forever and we still have that occassional brain lag where we just can't remember to unclip!
Life should NOT be a journey to the grave with the intention of arriving safely in an attractive and well preserved body, but rather to skid in sideways, champagne in one hand, strawberries in the other, body thoroughly used up, totally worn out and screaming: "Yeah Baby! What a Ride!"