OK, while I am not the most coordinated person in the world, I will try and comment. First, you don't necessarily have to have a dominant leg to clip in/out of. Just pick one and practice with that one. A lot of people think you should be comfortable/able to clip in and out from both sides, even if one is preferable. I am left leg dominant, but I started out as right. I switched somewhere along the way. You could go to a bike shop, have them set you up on a trainer, and see what happen when you are pedaling on the trainer, i.e. which leg do you favor when unclipping, when there is no traffic to worry about.
Are the pedals you are using easy enough for you to get out of? If it feels difficult or awkward, that increases the fear. Are the pedals one or two sided? Personally, I went from platforms, to spds (mountain style), to Speedplay Xs. I will give a big shout out to Speedplay products. So easy to get in/out of, double sided. The newer version is called Speedplay Light Action.

You need to practice in a safe place, like a parking lot on a Sunday morning. You can lower your saddle to start, to get used to the feeling, or try it where it is. Which brings me to ask, have you had a real bike fitting? You need to make sure your bike is actually set up correctly.

We've all had the bike shorts thing happen. Some shorts don't work with some saddles/geometry. It's purely a trial and error thing.
Stand over your bike, by approaching from the left side, swing your right leg over the top tube. You should be standing almost on your tiptoes, on both feet. Get a feel for this position. Clip in on the side you want to start pedaling on. Experiment with leaning the bike toward the clipped in side, with your other foot still on the ground. With the clipped in foot at the 3 o'clock position, push down with the clipped in foot and at the same time, push yourself up onto the saddle. You can just rest your unlippped foot on top of the pedal and don't get freaked out about clipping in right away. It might take a pedal stroke or two, or three! Practice riding around around for a bit and then you will reverse the process.
Unclip the foot you want to put down first, which should be the same one you clipped in with, in anticipation of a stop. It doesn't have to be way in advance when you are practicing, you decide. When you feel comfortable, slow down gradually, rest your foot on the pedal, and then slowly, lean a bit on the side you are unclipped on and brake and put the foot down. You should start out doing this at slower speeds, and eventually it will become automatic. You need to have enough practice for it to become muscle memory.
I learned this when my son took me out at 6 AM on a Sunday. I rode around for about 3 weeks, with one foot clipped in and the other not, in safe neighborhoods. Finally, I said, this is stupid! And, I never looked back. Sure, I've had a few slow speed falls, as everyone has.
It's important that you make sure your bike is correctly set up; my custom bike has a different geometry than my other bike, and frankly, it is more difficult to start on than my Trek Silque. No problem stopping, but the geometry requires me to to remember to really lift my body onto the saddle, where on my Silque, it is perfect.
So bike fit, pedals, shorts, and practice.