As Ed stated, you do not have time for both countries, even with 12 days. The flight down is a minimum of 20 hours from Seattle, with most flights going through LAX, although there are other options as well. Travel between Australia and NZ would take most of a day. Travel between the North and South Islands of NZ is also time-consuming, especially if you take the scenic route (by ferry to Picton). So you really need to pick either Oz, or the North Island, or the South Island. (The direct flights from LAX take you to Auckland on the North Island so most people start there.)
You should read up on the differences between the two NZ islands and between those and Australia to make an informed choice. Your kids would probably really enjoy Sydney and the local beaches (Bondi), and you could easily spend the entire time around there plus a few days somewhere else, Melbourne, Cairns or Brisbane (fly open jaw). They would also love the South Island around Queenstown ("adventure central"), and the wild west coast. The north part of the South Island offers beautiful beaches for walking and kayaking at Abel Tasman NP, and the Queen Charlotte Track near Picton offers more kayaking as well as mountain biking and hiking. The North Island offers more volcanic and thermal spots, Maori culture, the vibrant city of Auckland, and the beautiful Bay of Islands. Take your pick.
But have you looked at airfare? I checked random dates for a 12-day trip, Dec. 18 to 30. Seattle to Sydney on Qantas is $2900 pp in economy. It is a few hundred $$ less if you go earlier, around Dec. 14. Air Canada (through Vancouver) seems to be about the same.
Seattle to Auckland on Air New Zealand is also high, like $2600-$2800. They did not have seats available on any internal flights for the dates I checked, but that could change.
You might be able to find package tours (air and hotel) at a good price, but Christmas vacation is über high season so they do not need to make the pricing attractive.