Searching the NPS Soldiers and Sailors System you will find there were three Confederate units with that designation. Crawford's was also in the Price
expedition. I could find nothing of the other unit.

George Martin

1st Regiment, Arkansas Cavalry (Dobbin's)

1st Regiment, Arkansas Cavalry

1st Regiment, Arkansas Cavalry (Crawford's)

1st (Crawford's) Cavalry Regiment was formed during the spring of 1863 as a battalion, but two additional companies were soon added to bring it up to regimental strength. It served in W.A. Crawford's and W.F. Slemon's Brigade and fought at Poison Spring and Marks' Mills, then was active in Price's Expedition to Missouri. The unit was included in the surrender of the Trans-Mississippi Department. Its field officers were Colonel William A. Crawford, Lieutenant Colonel D. L. Kilgore, and Major J. W. Walker.