Phil - for what it's worth...I ran a check through Ancestry.com with this result:

U.S. Civil War Soldiers, 1861-1865
about A. J. Morrow
Name: A. J. Morrow
Side: Confederate
Regiment State/Origin: Texas
Regiment Name: 12 Texas Cavalry
Regiment Name Expanded: 12th Regiment, Texas Cavalry (Parson's Mounted Volunteers)
Company: G
Rank In: Private
Rank In Expanded: Private
Rank Out: Private
Rank Out Expanded: Private
Film Number: M227 roll 26

Cross-checked NP CWS&SS

12th Regiment, Texas Cavalry (Parson's Mounted Volunteers)
12th Cavalry Regiment was organized with about 940 men in August, 1861, by Colonel W.H. Parsons. Most of the men were from Hempstead, Fairfield, Georgetown, and Waxahachie, and Ellis and Hill counties. This unit served in Hawes' and Steele's Brigade, Trans-Mississippi Department and skirmished the Federals in Arkansas and Louisiana. During 1865 it was in Northern Texas guarding approaches from the Indian Territory. The regiment was included in the surrender on June 2. Its commanders were Colonel William H. Parsans, Lieutenant Colonels Andrew B. Burleson and John W. Mullen, and Majors Locklin J. Farrar and E. W. Rogers.

A. J. Morrow
Regiment Name 12th Texas Cavalry
Side Confederate
Company G
Soldier's Rank_In Private
Soldier's Rank_Out Private
Film Number M227 roll 26

J. C. S. Morrow
Regiment Name 12th Texas Cavalry
Side Confederate
Company I,H
Soldier's Rank_In Lieutenant
Soldier's Rank_Out Captain
Film Number M227 roll 26

IF A.J. Morrow is related to Capt J.C.S. Morrow, that might explain a few things. Sifakis says unit was also known as the 4th Texas Mounted Dragoons. I'm wondering if -- maybe -- he just switched units.

U.S. Civil War Soldiers, 1861-1865
about A. J. Morrow
Name: A. J. Morrow
Side: Confederate
Regiment State/Origin: Texas
Regiment Name: 13 Texas Volunteers
Regiment Name Expanded: 13th Regiment, Texas Volunteers
Company: F
Rank In: Private
Rank In Expanded: Private
Rank Out: Private
Rank Out Expanded: Private
Film Number: M227 roll 26

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.

I checked Census records. Is this your man?

1880 United States Federal Census
about A. J. Morrow
Name: A. J. Morrow
Home in 1880: Precinct 2, Lavaca, Texas
Age: 52
Estimated birth year: abt 1828
Birthplace: Alabama
Relation to head-of-household: Self (Head)
Father's birthplace: Alabama
Mother's birthplace: Alabama
Neighbors: View others on page
Occupation: No Occupation
Marital Status: Single
Race: White
Gender: Male

Andrew J Morrow
Born: 16 Dec 1845
Henderson, Texas, USA
Died: 23 Feb 1933
Spouse 1
Augusta
Born:
Died:
Marriage:
Children Sex Birth

Spouse 2
Mahala
Born: 28 Jul 1850 in Texas, USA
Died: 20 Feb 1882
Marriage:
Children Sex Birth
Julian Morrow M 1876 in Texas, USA
Meta Morrow F 1877 in Texas, USA
Alma L Morrow F Mar 1878 in Texas, USA
Eva M Morris F Jan 1880 in Texas, USA
Hallie E Morrow F 1882 in Texas, USA
Andrew J Morrow M Sep 1885 in Texas, USA
Benton P Morrow M Jan 1888 in Texas, USA
Nestor Morrow M 23 May 1895 in Texas, USA

Spouse 3
Mary Isabelle Berry
Born: 1847
Died: 1929
Marriage:
Children Sex Birth
Eva M Morris F Jan, 1880 Texas, USA