You're getting good advice here and it's a plus you have two bikes to ride on because if you only had the one that you feel less stable on you wouldn't be out practicing. It sounds like you have some level of comfort and are ready to practice with your road bike. Like Alice said, take it slow and don't rush yourself.

It is embarassing to fall, however I care more about the danger of falling, like breaking bones or otherwise seriously hurting myself than the embarassment of it. I had my share of falls last year when I was starting out. It was a combination of not being acclimated to a bike from years off of one and a bike that was too big for me. A trip to your local bike shop requesting that someone observe you riding can help to dial in your bike like seat and handlebar adjustments and then you can rule out fit. Riding, riding and more riding will gain you confidence and comfort on your bike but it sounds like you already have that.

And welcome to TE!