Unfortunately, there are no muster rolls for any of the companies of the 34th Texas Cavalry which precludes knowledge of soldiers service. The two requisitions recorded in his file are for another J. M. Shivers, not your man.

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J. W. Shivers, 2nd Lieutenant, Company A, 34th (Caudle's) Texas Cavalry, Alexander's Regiment, 2nd Partisan Rangers, recorded on a list of commissioned officers elected, and those relieved at the re-organization, June 27, 1862, no other records

M323: Compiled Service Records of Confederate Soldiers Who Served in Organizations from the State of Texas

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34th Regiment, Texas Cavalry (Alexander's) (2nd Texas Partisan Rangers)

34th Cavalry Regiment [also called 2nd Texas Partisan Rangers] was assembled and accepted into Confederate service at Tahlequah, Indian Territory, in the spring of 1862. The unit served in Polignac's, King's, and J. E. Harrison's Brigade, Trans-Mississippi Department, and fought at Spring River, Newtonia, Prairie Grove, Mansfield, Pleasant Hill, and Yellow Bayou. It was included in the surrender on June 2, 1865. The field officers were Colonels William M. Bush, John R. Russell, and George H. Wooten, and Majors Thomas J. Dove, M.W. Davenport, and Sevier Tackett.
http://www.civilwar.nps.gov/cwss/soldiers.cfm

See also:

http://gen.1starnet.com/civilwar/34cvmain.htm
http://www.tshaonline.org/handbook/online/articles/qkt34
http://www.tshaonline.org/handbook/online/articles/fal06
http://www.texansinthecivilwar.com/34th_Cavalry/company_a.html
http://www.pricecamp.org/campancestors/james_barrington_davis.htm