I biked across Oklahoma from Black Mesa east 101 years ago - early October - perfect time of year.
Took the GRE at OPSU - slept by the ball fields. Also visited the music faculty at Northwestern in Alva.
To this day, I had the best tailwinds of any trip - - ever - - stretched out my arms and went 20 mph.

A note about random camping in the Great Plains states:
Most of the states permit "roadside" resting and the counties own right of way on county roads.
If you ride 1/2 mile or so off of a paved road - maybe a mile -
You can usually find a nice spot where farm roads intersect with an area to the side that is semi-clear.
If you wait to evening - there will rarely be any traffic to speak of until morning - but tractors will be out early.

Nice write-up of your trip - - J