OK, here goes an example:
Take I-84 to intersection with I-80 E.
to intersection with I-25 S.
to US 87 in Raton, NM
In Amarillo take US 287 to near Wichita Falls
Take US 82 E. @ Henrietta, TX to Texarkana
In Texarkana take I-49 South to Louisiana
I believe this would be the fastest route.
Small adjustments could take you by Yellowstone, and a major adjustment could take you by the Utah and Southwest Colorado sites.
Another small adjustment on the original route could take you by Santa Fe and Northern NM sites.
US 82 avoids the congestion of Dallas/Fort Worth.