Lol, I read your post and new exactly what you did. Former traveling nurse speaking here. BTW, I have freinds traveling right now and they are making a killing.
Since you can write of your mileage (assuming you own a home), a lot of your expenses will be defered gas-wise anyway. The dogs are really the complicating factor.
Three pragmatic factors come to my mind are:
1) Size, you need space for you dogs.
2) Traction, a lot of popular travel destinations are in areas known for snow. Colorado, Maine, Northern MI (Traverse City), or here in the mountains of NC. Alaska also has asignements though most folks try and go in the summer.
3) Towing Capacity, the number of travelors I have known that have needed to tow is more than a few. They buy furniture or something they need to take "home." The other thing I have seen is that they meet someone special, if you know what I mean, and end up wanting to tow one of the vehicles behind the other.
The vehicle that comes to my mind, and I don't think it has been suggested is a Nissan Armada. Subarus are definately indigenous to mountainous regions where the snow flies and the Tribeca's are impressive space wise but lack serious towing ability.
Frankly, if I were to do it again, I would take the houseing subsidy. I would get an RV and pull my vehicle behind it. You could probably make 3 RV payments for every month you worked.
Good luck man in your travels. And if I may suggest checking out
www.missionhospitals.org. A very popular destination with traveling nurses.