From your description I would guess that your problem not about learning how to do it (you can do it, but only on one side) but more about fear and confidence.
There are two things you will need:
- some quiet spot to practice with smooth surface and not many obstacles, maybe a parking ground or so,
- a bike that has a preferably upright riding position.
The reason for the latter is, that your hands on the handle bar serve two purposes: the hold the bar to control the front wheel and they support your upper body. The bent over your upper body is on the bike, the more weight has to be taken by your arms to support your body and the more can your upper body influence the steering behavior (as already mentioned in webbgeek's answer). With an upright position the weight of your upper body is lasting on the saddle via your hip and the arms only have to steer.
So if you have different bikes at hand, chose the one with the most upright position you can get – chances are good that this is your grandma's bike or something similar ;-) If you have no other bike but different possibilities to place your hands on the handle bar, start your practice with the hand position that allows for the most upright position instead.
When on your safe spot now, try to ride some straight lines and circles while putting as less weight as possible on the handle bar. Also try not to wrap your fingers around the bar but only put your palms or the joints between hand and fingers on the handles while keeping your hands open, so that you have only the friction between your palms and the handles to control the steering. By trying to put as less weight as possible on the handlebar you also will sit more upright which will make it easier to use your upper body and hips for steering.
You will also realize, that you don't have to hold the handles tight like you tried to squeeze water out of them. Once you feel safe with this situation you can now try to take away the weight from one hand completely without putting this weight on the other hand and also without closing the remaining hand to fasten its grip. Maybe it helps, if you don't take away the weight by just lifting your hand but by rotating your shoulders a bit, so that the shoulder of that side, where you want to take the hand away, goes slightly away from the handle bar. Now you can again first practice going straight and then some (hopefully now intentional) wiggly lines which should give you more safety while riding with one hand.
I also would recommend to do this not only with your weak hand, but practice with both hands and watch if you unintentionally fall into some certain behavior while riding with your strong hand.