There's a website with safe (ish) bike paths across the country. The one you might be interested in is the southern tier
My route from San Diego to Slab city went something like this:
Hwy78 from Escondido via Julian and then to Brawley. From Brawley I went to Slab city (on the 111 obviously).
In Arizona you're allowed to ride on freeways, so my route was like this:
From Slab city I continued on the 78 to Blythe, got on i40 until 60 and made my way to Prescott, Sedona.. you'll see on the map, it's pretty straight forward (60, then 89). Then to Flagstaff on the 89A.
From Flagstaff you can ride the i40 to ABQ and then take the torquise trail to Santa Fe, or you can from Flagstaff go a bit north and take the 160, but it's a bit of a detour.
Anyhow, from Santa Fe I went north (I aimed to Chicago) so I took mostly the 285 through Colorado to Denver. No freeway needed there.
Houston:
I was supposed to bike from New Orleans to Houston on the Galveston bay road (first to Laffayette and from there south, hwy 82 to Port Arthur) but due to floods and road closures I got a ride from Laffayette to Kemah. from Kemah I used highway 3 on a Sunday, so it was chill to Houston. But definitely not a fun highway on a weekday.
From Houston to Austin I used the 36 and then 71, there's no other way.. 71 is horrible. Lots if construction and weird interchanges make it annoying to bike (on the map it seems easy. It ain't that obvious in real life).
Aggressive drivers are annoying but I mean, the real danger is when you get into a city, because usually when it's the highway you'll have wide shoulders..