The route you are taking is almost the exact same route I traveled on my honeymoon many years ago with the exception that I started out in Washington DC and ended up in Portland, OR. I also was traveling in winter time, we got married on February 24th so most of this trip was in either late February or early March.
When we stopped in New Orleans, a local recommended that we take the St. Charles Streetcar Route for some inexpensive entertainment. Back in 1974, it cost a nickel to ride this streetcar, I'm sure it's a lot more nowadays! It gets you out of the French Quarter and passes by mansions, restaurants and Tulane and Loyola Universities. You can also get a pass so if you want to get out and capture images, you won't have to pay the fare again and again.
http://www.experienceneworleans.com/new-orleans-streetcars.html
I've been to Austin (although it was on a completely different and more recent trip) and can highly recommend a restaurant with stunning views (known for its sunset views) of Lake Travis. It's called The Oasis. Here's a link which gives you an idea of the kind of images you can get by going to The Oasis at sunset:
http://oasis-austin.com/sunset-photo-contest/
You knows, you could even win the contest.
It's not in downtown Austin, you'll have to drive about half an hour to get to the the restaurant. Well worth the trip.
The absolute highlight of the trip, and one that counts as one of my all time best life experiences was the mule trip into the Grand Canyon. People make reservations a year in advance to do this trip in the summer, but, because we arrived in early March, were able to do the trip without any reservations.
Because we were limited in time and money, we took the day trip down to Tonto Plateau. If we'd had more time, we couldn't done the overnight trip all the way down to the Colorado River. Be prepared to shed your heavy winter coat as you head down to lower elevations. I still talk about this trip with great enthusiasm even after all these years. If you have the time, I highly recommend it to enhance your Grand Canyon experience.
Enjoy your trip!
Ellen