I was exactly the same, and yes it was difficult. Maybe just like Frasier my brother turned out to get it right first, so I kept on trying, when I would ordinarily have given up.
As Megaten suggested you could try training wheels. I don't know if that's possible with the bikes you won. I do think however that if I showed less concern when learning I would have mastered it earlier. I know people with anxiety find that hard, but just try to convince yourself it's no big deal whether I get it right now, or whether I get it right in a year from now. I think that's probably what did help, realising that as long as I kept at it, it was impossible for me not to get it right, and that proved to be correct.
Also, don't be embarrassed. If you were a kid, you wouldn't be embarrassed. You wouldn't expect a kid to get it right without any difficulty. So you're the same, only you're learning to ride at a different stage in your life. That's nothing to be embarrassed about. It's not that you're incapable of riding a bike, you just didn't learn at the same time everyone else did.