It's not the unclipping, because there are people who do the same thing on motorcycles. Cleats just add another complication.

If I were you, I'd put the flat pedals back on for now, with no clips either, until you've learned how to stop while remaining in control of your bike. Then practice, practice, practice.

One thing I didn't see anyone else mention here is initiating the lean *before* you stop. (Apologies if I missed it in skimming over this thread.) That's a critically important part of the sequence, but one that people who've been doing it forever tend to forget about. As you approach your stopping point, push gently forward on the handlebar on the same side of the foot you'll be putting down. That makes the bike start to lean toward the side that you pushed on. Now you're secure and you know which side your bike is leaning toward, and that it's going to be the side you're preparing to put your foot down.

Practicing with a curb as Geonz suggested is one way - but please don't get into the habit of using a curb when you're on the road. If you're that close to the curb when you're riding, you're putting yourself in a position to get buzzed, rear-ended or hit by a mirror while you're on the straights, and right-hooked at intersections.