Tons of good advice here, can't add to it by much. Over the years I have focused on learning to peddle in a way that does not impact my steering (wobble). This demonstrates that I know how to ride a bike straight and car drivers don't get as nervous because you look predictable. Same is true for bikers passing you. It takes time but it actually is a great help in a number of ways including improved efficiency.

Double check the fit of the bike. If your seat is too low, you will not be as steady, which is true if it is too high. Place your feet in a comfortable position on the pedals which goes along with bike fit. If you are comfortable on the bike, you can focus on the driving.

Keep up the good work! I have started again after a nearly a year of no riding and average about 12 mph. It's OK, it will only get better. On this note, make sure your tires are inflated to the proper PSI. If they are low, you will be slow and exert a lot of energy.