Yes, you can do everything you want. In order to get the renewal rate on your AP you need to renew during a window that opens 30 days before the AP expires and closes 30 days after it expires. You'll be well within that renewal window, so you'll have no problem getting the renewal rate (and you'll qualify for the DVC AP since you're a DVC member as well).
You do need to redeem the voucher and do the AP renewal in person at the parks. You'll need to go to the will call window at one of the parks and redeem your GAD vouchers (everyone with a voucher will need to be present and show ID, including the kids). Then tell them you want to use it to renew your annual passes. They'll give you credit for an adult one day ticket toward the DVC renewal price, so you'll pay $282.22 (including tax) for the adults and $251.34 for the kids.
Also note that Disney has a promotion going on now that will give you 15 months if you renew an annual pass before August 31st. Since you'll get in under that deadline as well your new AP's will expire 15 months from the current expiration date.