Not sure how much of Route 66 is left standing, much of it in New Mexico parallels I-40 and in places it maybe replaced with I-40, where it remains I would think it would be safe. Across Texas and Oklahoma I believe it is much the same but not all of it will be following I-40 in OK. There are signs off the interstate saying where sections of 66 are still around in NM and OK. In Albuquerque it runs through but often in seedy areas, but there are Route 66 museums in other places in OK and the old gas stations, cafes and motels in both NM and OK and I assume other states.
Are you planning on dropping down to Douglas at Albuquerque or over in AZ. The shortest route would be Albuquerque to Hatch, then take SR26 which is a short cut to Deming and on to Lordsburg; past Lordsburg take SR80 at exit 5 @ Road Forks for Radio and on to Douglas.
The Interstates are always safer in my mind; no oncoming traffic, no stop lights
, noone pulling out from side roads, but crazes are out there, whereever, always be alert.
Have a safe trip and a happy one. Welcome to the Southwest. This should be a good year for the wildflowers blooming.