The problem starts with how you mounted and centered the caliper.

Dual pivot brakes must me mounted square and level, after which the centering is fine tuned with the balancing screw (on top of the arch).

Note, I said fine tuned.

What happened, is that the screw was used to throw the two caliper arms out of balance, than the caliper was centered based on that.

Start by removing the cable and both brake shoes. Loosen the pivot bolt, and rotate the caliper so both arms are equally long. You'll need to adjust the balancing screw and with a minute or two you'll find you can square it all up, so the arms are equally long and centered. Tighten the pivot bolt to hold it there, string the cable, mount the shoes so they touch the rim about 1/16" below the upper edge (so the won't touch the tire after there's some wear.

When finished, adjust the opening, and fin tune the centering with the balancing screw.

Hint: squeeze the caliper closed with your hand to take the load of the screw when you adjust it.