I'm so glad you posted this. I'm 50+ and returned to cycling in March. I'm pretty much at your level of riding, 20+ miles on weekends, about 10 miles a few nights of the week. I just can't seem to get rid of the "wobblies" even with both hands on the bar. Removing one hand guarantees a swerve in one direction or the other.

I'm glad to know that more practice and strengthening my core will help. I ride a hybrid so I ride more upright, but still....