First of all, check that the lever on the back brake shoe is not seized. It's riveted to the shoe and is supposed to pivot round its rivet then the cable pulls it. On my daughter's car, the handbrake not working on one side was partially down to that.
Slacken the cable off fully at the handbrake lever end. You might find the auto adjusters are seized on the rear shoes. They often are. You'll need to pull the drum off and flick the little knurled adjustment wheel round with a screwdriver. Keep doing this and replacing the drum until you can JUST feel it starting to drag. It's a pain-in-the-backside process and if there is any kind of lip on the inside of the drum, you'll struggle to get the drum off once the adjustment is close to being correct. The handbrake has such leverage that it can only work effectively with very TINY movements of the shoes. Each time you replace the drum, you'll have to spin it a few times and jab the brake pedal to centralise the shoes. They really need to be almost dragging for the whole thing to work.
Make sure the new cable is pushed fully home into the abutment hole in the backplate. Ours had a bit of "spring" to it and when the handbrake lever was pulled, it just pushed itself further itno the abutment hole and lost some movement.
Tighten up the cable adjusters at the handbrake end LAST. Take as much slack out as you can without the rear brakes binding.
