I did locate a "J.F.Carter" listed in the 23rd Va Battn Co.H (name is also sometimes listed (copied) as J.T.Carter... A common mistake found in post war archive card copies of records since earlier style of writing an "F" "T" "J" "I" can easily be misinterpreted to the untrained eye.
Company H of the 23rd Va Battn was a mix of soldiers that some had previous service in the 2nd NC Battn, and some that had come from the Virginia State Line troops (state service units) Unfortunately the Va State Line units didnt last too long in existance, and what records that might have existed were mostly destroyed at the end of the war in the Richmond Fire, so what remains is fragmented.
The 23rd Va Battn mostly saw service in western Va and participated in the engagements at Greenbriar River and Droop Mountain. In 1864 it participated in the Shenandoah Valley Campaign, then simply disbanded in April 1865.
Respects
Frederick