I don't like correcting with external aids because they change the skill unpredictably.
Maybe you can incorporate some of these thoughts into the work you do with your coach.
First I want you to understand it's my impression this is not a new skill, and the problem you've described is a chronic one rather than the usual whoops that happens while learning any skill. I also want to say these suggestions can be applied to anyone they stick to because I'm addressing common issues shared by many gymnasts..... so it's not you, it's me.
Sure, you gotta experience change to go from sorta to really good, but I want that change to come from you. Really it comes down to first changing yourself to image the correct model of what the bhs is, and then repeat the skill over and over in your imagination to the point of anticipating what you'll feel, hear, and maybe even see at key moments of the skill. Make it as real as you can, and do it all day long right up to the final moments (especially) before you fall asleep at night.
Imaging to this extent isn't some feel good comforter to build up your confidence, because that's a shallow process. Your mind and body are both made of the same stuff, and work together all day long to perform simple and complex movement, so working your mind will add to what your body has learned and make the skill easier. I guess you could say the concept is similar to using your fingers, as a child, to keep track while counting to ten.
The next step is to work on the floor line and do the skill within a very narrow description of perfection. Every attempt is going to receive your total attention and energy to allow your version of perfection to be the result. When you do these you'll learn to have such control that there's nothing but intended movement through to the finish. These will finish balanced, with square hips, and straight alignment that will allow you to land the second foot and after the briefest pause move to a skill on either your front or back foot. Repeat this entire process and make it a bit briefer on each successive increase in beam height.
That's the confidence building control you're looking for, and right now you don't got it, but you really can have it in very short order by cramming a bunch of floor line bhs into any amount of time you can sneak them in...... or let your coach know you want to turn over a new leaf and need 3 minutes each day, which is enough time for twenty on the line, or floor beam 18 on a low beam and 15 on the high beam..... just slowing enough to allow time for the occasional fall and climb back up. No, you don't do the line and the floor beam and the low beam and the high beam with those numbers in a single workout. Warm up for the beam height you want to use, and then get your numbers in.
Meanwhile consider this line of thought. You go to the gym each day to get a little better, but have struggled for too long on this skill. You may have had a genuine reason to struggle six months or a year ago, but not now because you've had time to get a little better every day. The problem isn't your lack of physical ability, but your sense of your ability to do the skill hasn't grown while your body trained and get better. Your sense of ability just stayed there, where it was the first few weeks or months you you trained them.
It's time to let your mind catch up.
Here's another something to read......
You say you get your hands on the beam, then get loose, then get your feet on, and sometimes stick and sometimes not. Getting loose is making the skill harder to do safely because, well you don't practice "loose" as if were a functioning part of a skill in the sense you strive to do it consistently and with polish it. So I'd say that "loose" is a wild card that may change your core's orientation to the beam and place your legs slightly here once time, and slightly there the next time, and some other place times 15. You'll never have a sense of security when you have 15 different versions of your body doing the second half of the skill because you can't train something that changes 5 times out of 10. Really, force your body to give your brain one predictable problem to solve.
Doing it right may seem to hard, but doing it wrong to make it easier will never work..... it'll make it harder.
Go now, and be good to yourself.
in that order.
