Katie.
NZ is a great country to tour just be careful of the drivers. Many are not so kind to cyclists.
Bike - you need a woman's specific frame whatever make and model you choose. Not sure if all manufacturers do WSD frames. Check out a few.
Dawes Galaxy or Horizon,
Roberts (Josie Dew's bike)
Condor (Ann Mustoe)
Thorn
Orbit
Or hardtail MTB with stronger 26" wheels mudguards and racks (might be better if you plan going off road or off the bitumen). Plus the lower gearing may be better for you than road bike gearing climbing the long hills loaded up especially if you get onto some of the forest tracks. I think an MTB 26" wheel based tourer would be best and strongest.
I'm sure there are more manufacturers, perhaps some one else such as Rich P can suggest a few more.
As has been stated above a budget would be most helpful.
Some of the bikes mentioned above such as the Koga Miyate are top touring bikes and mega expensive. I would suggest a 2nd hand tourer might be a better option if you are new to touring and only going for a month. Also budget for decent racks if the touring bike you are buying strangely doesn't have racks. The best racks are Tubus. The best panniers are Vaude, then Ortlieb, then the rest. Try and keep weight to a minimum as I 'm sure you will realise.
How about buying a bike new or second hand or hiring one in New Zealand? This would save you having to pack one up to fly with and the extra cost as well. Just suggestions.