1. you'll get better with time. the anxiety is understandable, but with practice, that will fade. spend time in an empty parking lot to avoid the stress of traffic and intersections.
2. most of us cannot put our foot down when in the saddle, and, frankly, you should not expect to do that. when it's time to stop, you stand on the pedals, usually with one foot low, and then put the opposite foot on the ground when the motion stops. Notice that this implies you can stand over the top bar when stopped. (if you can't do that, you do likely need a smaller bike).
3. for a "start" to checking the fit, you should have a friend hold your bike while you're on the seat, still, and you should put your heel on the pedal. Your leg should be close to straight, not reaching, not significantly flexed at the knee - then your seat is "in the ballpark".
4. I'm confused that you're saying "hit the brakes HARD to stop." Brakes should feel pretty relaxed when they're doing their job, else they need adjustment. You need a local bike shop (LBS) or a knowledgeable friend to take a look at those. Something isn't right there.
5. Sorry, but 5'1" does not get you into the "super short" category. I've ridden with some much more vertically challenged than that.
6. spend some time with an online "fit calculator" and see if you think you need a different sized bike. the calculators aren't gospel, but they (again) get you in the ballpark. try this one:
https://www.jensonusa.com/bike-fit-calculator
7. be optimistic. you're not as far from bike-happy as you might think.