Just like everybody else said, there is no need to quit. I am quite prone to mental blocks and the first few really bad ones I thought that I wanted to wuite, but once I got over the issues I loved being in the gym again.
For back walkovers beam start with the low beam and work your way up to the medium beam, then go to the high beam with the mats stacked under. (When I had mental issues with backhandsprings I used a lot of little mats so I could take them out easily.) Once you can do ten in a row without balking then move the next mat away. Once you are on you last few mats ask a coach to spot or stand there while you take away the last few mats. This is how I got over the fear of backhandsprings on the high beam. For the backhandsprings with the mats. JUST GO! (I don't mean to be harsh or anything but I know what you are dealing with and sometimes tough love, really works (personal experience). You will be ok and sometimes counting 1, 2, 3 might help. If this doesn't, ask a coach to stand there and have, him/her slowly back away, until its all you.
For flyaways, start on the low bar with a spot (even if you can already do a flyaway, try it. It might be good to back track a little, when you are mental.) If you feel fine after a few do some on the low bar without a spot until you a confident. Then put the high bar as low as it can go and do it off there with a spot, then without a spot. Try raising the bar a notch then do the same thing. Keep raising it until you get to where the high bar is usually set. If you ever get to a step where you can no longer do it then back track one step until you are comfortable then move on again.
Always ask a coach for help when going through a mental fear. Eventhough each person, usually solves mental bloks different ways your coaches can support you and maybe have other ideas on how to get past it because they know what kind of person you are and what might help. Try to ask your coach for a private lesson too. For me it was way easier to come in with my one coach then have the all the optionals staring at me while I tried to get skills back and work through mental issues.
4 more important things to remember though:
1. each person is different. What might have worked for a friend may not work for you and thats ok. You need to find a way that you solve these fears.
2. A little fear is ok. I hate to tell you but, you will always be afraid of different skills Some will be easier then others. You need to learn how to not let the fear get the best of you and cause a mental issue, instead try to get rid of the fear of that skill by doing it a lot.
3. A little 'tough love' is sometimes a good thing thing. After about almost a whole season of me being afraid of jumping to the high bar (I am a level 8 so this is pretty bad, haha), my coaches got kinda mad at me and one of them even told me to quit in a round about way and almost told me I couldn't compete etc. Well a month later I completely regained the skill, so if the coahes are being a little harsh don't let it get to you too much.
4. There is no set length of time that it takes to get over a mental block. It may take a season or it may take one weekend. If you don't get something right away, keep working at it because if you put your mind to it you will get over it. Just keep working hard and if its ok with you coaches ask to stay a few minutes after practice or come a bit early to work on it. Plus this shows them you care about all this and when I had issues they always knew it wasn't on purpose or to get attention or anything and it shows them you still deserve to be a part of the team even if you don't have all your skills right at the moment.
Sorry this is kinda long, but I hoped some of it helped. Just work hard, believe in yourself, don't give up, and SMILE

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Good Luck!