Could it be that you're spinning in too low of a gear? When I'm going really slow (like riding with my kids) I feel this sometimes unless I shift to a slightly harder gear. I find it's easier to go slow pedaling at a slower cadence than to try to spin fast while going slow... there's little/no resistance during parts of the stroke at super low speeds and it's kind of unnerving.

I hadn't been on a bike in quite a while before I bought mine in May of this year.. and I had clipless pedals by September and wished I had gotten them much sooner. I have yet to fall, but I don't expect that I won't eventually - I think falls can happen, clipped in or not. Since you already know how to clip in and out I wouldn't be afraid to see if that helps you on your bike