Sherry,
First, welcome to the ACA Forums! Second, welcome to bicycle touring. WARNING: It can be a very addictive passion.
Regarding your question. You could ride east on NM-6 and then north on NM-47. However, once you get on NM-47, the shoulder is minimal and there is still a lot of traffic. It is legal and safe to ride on the Interstate in this area (follow ACA's routing as you get closer to the metro area). There is a full-width shoulder in addition to a rumble strip at the right side of the outer lane to warn vehicles they are drifting into the shoulder so it would be quite unusual for a vehicle to drift onto the shoulder and hit you. Plus, the road grades tend to be a little gentler if not longer and the sight lines are generally much better so drivers can see you easier.
The big cons are Interstates are typically much noisier, you have a high chance of a flat tire due to little wires from shredded truck tires (avoid them!), no shade if it is hot, no trees to block the wind (can also be a positive), and the scenery is typically blah.
I have LOTS of touring experience. While I do not enjoy Interstates generally due to the cons, I will ride them if needed. In this particular area, I would personally ride the Interstate with no reservations (other than shredded truck tires) versus the above alternative.
As an FYI, you are generally allowed to ride the shoulder of Interstates WEST of the Mississippi River outside of metro areas if there is no "reasonable" alternative like a paved frontage road. In this case, there is no "reasonable" alternative as the above alternate I listed would add almost 20 miles or 2 hours of riding.
Hope this helps and I wish you a wonderful tour!
Tailwinds, John