If you can't get in at the state park, one area of Amarillo that is OK for a one night stop over would be I40 at Coulter/Soncy- has your basic chain type hotels- Courtyard, Comfort Suites, Holiday Inn Express kind of thing.
Funny how many people I know who are either moving to NM or have a 2nd place there- my sister bought a townhouse in Red River last year, friends just moved to Las Vegas, NM, another couple have had a cabin Cloudcroft for years. Now a lady I work with is contemplating Santa Fe.
GUess we will have to skip a Sowal vacation someyear and head west one state!