This appears to be the same William Woods in Brown's Company, 13th Texas Cavalry (Burnett's), age 22, enrolled at Homer on March 22, 1862 for 12 months. His service record says "Deserted Homer Angl Co 15 Apr 1862". In another place it says "deserted May 5".
In the same regiment is a Henry Woods, also from Homer, Angelina Cty, age 42 who is discharged due to his age under the Conscript Act.
Many times "deserters" show up in other regiments, as does William.
On May 9, 1862, William Wood enrolls at Homer in the 22nd Texas Infantry (Hubbard's), company D (Anderson's), age 23.
I didn't find a record of his death.
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13th Regiment, Texas Cavalry (Burnett's) (13th Mounted Volunteers)
13th Cavalry Regiment was organized at Crockett, Texas, during the winter of 1861-1862 with about 900 men. Some of its members were from Centerville, Crockett, Madisonville. In the summer of 1862 it was dismounted and later assigned to O. Young's and Waul's Brigade in the Trans-Mississippi Department. The unit participated in various conflicts in Louisiana and Arkansas including the engagement at Jenkins' Ferry. It disbanded during the late spring of 1865. The field officers were Colonels John H. Burnett and Anderson F. Crawford, Lieutenant Colonel Charles R. Beaty, and Major Elias T. Steale.
22nd Regiment, Texas Infantry (Hubbard's)
22nd Infantry Regiment was organized during the early summer of 1862 using the 5th (Hubbard's) Texas Infantry Battalion as its nucleus. Many of its members were from Palestine, Quitman, Livingston, Mason, and Tyler. The 22nd was attached to O. Young's and Waul's Brigade, Trans-Mississippi Department, and fought in Louisiana and Arkansas including the conflicts at Young's Point and Jenkins' Ferry. Later it moved to Shreveport, Louisiana, then Hempstead, Texas. Here the regiment disbanded before the surrender in June, 1865. Its commanders were Colonel Richard B. Hubbard, Lieutenant Colonels John J. Canon and Elias E. Lott, and Major Benjamin F. Parkes.