I tried all the spellings you listed in Booth's Record of Louisiana Confederate Soldiers and this is the only match I found...

Foulks, W. C., Pvt. Co. H, 28th (Gray's) La. Inf. En. Monroe, La., Sept. 13, 1862. Present on
Roll for July and Aug, 1863.

I did find a John Faulk in the 1st Battalion, Louisiana Cavalry Scouts, a Union unit that operated in that area. Is this possibly kin to your men?