Don't give up yet-- I had a handful of clipless falls starting out this January. They were mostly to do with not appreciating my center of gravity while clipped in on one side. That was a real downer, but I got my confidence back and really like them now. Here's what helped me:
When you're leaving one side clipped in, which is what I do the whole time, you have to deliberately lean the bike to the other side. It's very easy.
When you're getting used to your pedal system, make sure that you unclip well ahead of your stopping point. You can leave it later as you get more proficient.
Always plan your gearing so that pushing off on one side you have a tall enough gear to get you some distance to clip in, but not too tall that you have trouble pushing off.
Just have some patience, it'll become second nature.