I've no experience of the Cavalier handbrake but this may help:
On Fords the handbrake lever will only pull a link cable which will connect to an 'equaliser' (like a semicircular pulley) and the main brake cable will wrap around this and go to each rear wheel. The principle is that the equaliser will be pulled forward by the lever and link cable, pulling the brake cable with it. When the shoes on one side contact the drum that half of the brake cable stops moving and the other half is pulled around the equaliser so that that side continues to apply and the lever can still be pulled. Once the shoes on both sides are in contact with the drum then the brake cable, equaliser and lever cannot be pulled any further.
Ideally the brake shoes on both sides will move at roughly the same rate and the friction on both sides will be about the same, therefore both brakes will apply at about the same time. However, if there is unequal friction on both sides of the brake cable then one side will apply before the other. The friction can occur in three places: 1) The equaliser itself may not be free to swivel around it's pivot with the link cable, or the main brake cable may not be free to slide around the equaliser. 2) The main cable may be sticking on one half, caused by contamination or fraying for example. 3) The shoes, operating lever or self-adjuster on one side may be stiffer than the other. This could also be caused by incorrect release spring fitment or missing release springs on the brake shoes.
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As you've already replaced both brake shoes and you mention the brake cable is fraying, I would suggest you replace the main brake cable before going any further. Whilst it is off you can compare the effort required to actuate each rear brake (ideally using a spring balance) to see if there is any unevenness there, and you can check/lubricate the equaliser mechanism.
Hopefully there should be some clues above which might help.
Darren