Keeping rim brakes centered
Showing a photo of the brake calipers would help, but from your description I'd check such things:
Make sure that brake cable runs in Bowden housing freely. You can try lubricating it, but if there is any friction, no matter if from worn inner sleeve or frayed cable, whole set should be replaced.
Second thing is to check springs on calipers and their mounts. Sometimes plastic housing breaks and do not keeps spring properly. In such case some replacement is needed. But first check those springs adjustment. There are small screws at the bottom of calipers to increase &quot;opening force&quot; of caliper.