The animal is too big to travel in the cabin and has to go in the hold.
I presume you are flying yourself, and the dog should accompany you.
The only airline with nonstop flights between NYC and Austin accepting dogs as checked baggage is American. JetBlue only allows smaller animals in the cabin, Delta and United do not accept dogs as checked baggage but make you use their cargo services ('PetSafe') at vastly higher prices, and Alaska's flight is actually operated by American .
The fee to fly your dog there on AA is $200 one way, which cannot be paid with miles/points.
View AAs animal transport policies here: https://www.aa.com/i18n/travel-info/...tance/pets.jsp

Tips:
- As AA strictly adheres to their published temperature policies and NY winters can get really cold, get an acclimatization certificate from your vet that covers the maximum permissible temperature (20 degrees).
- Allow for flexibility in case of really low temperatures. AA will rebook with no fees if the dog can't travel on a given day.
There is just one nonstop a day (Flight 2675), which leaves JFK at 5:30 pm. Of course, there's lots of connecting itineraries, but nonstops are definitely better with animals in the hold.
- Be at the airport early. Policy used to be 120 minutes prior to departure, and although the wording on the website has been changed to a generic "allow extra check-in time" about a year ago, staff will scoff at you when getting there later than 120 minutes prior to departure.