As a "New Orleanian" of sorts (though I live here and my family is from here, I wasn't raised here), were I planning to do the ride you are considering, I'd head south/southweastward, hook up with the ACA Trans Am route/Katy Trail (across Missouri if I had tires wider than 28's), head east to hook up with the Mississippi River Trail (St. Louis), and then follow it south to New Orleans.

Some advantages:
1) well-established & documented routes (the TA and MRT are documented enough via online journals that you might be able to forego buying the official maps given enough online time; then again, you could buy the "official" maps/sections/books.)
2) fairly difficult to get lost even soloing
3) even the locals are familiar with "cycle touring" to some extent (a plus as you might be more likely to have warm showers type hosts/free public camp spots).

A few possible disadvantages:
1) longer than simply cutting diagonally/directly SE
2) weather - either a late, cold winter as you leave Colorado (longer route means leaving a week or so earlier) or an early heat wave as you descend into Louisiana