The best short history of each La unit is Art Bergeron's Guide to Louisiana Confederate Military Units.
The first Regulars or Strawbridge's regiment started the war at Pensacola FL, then were transferred to the Army of Tennessee in time for Shiloh and generally stayed with the AOT for the remainder of the war. Strawbridge started as a company commander and did not get the regiment until early 1863.
The First Regiment of Louisiana Volunteers were sent to Virginia and served in the First LA Brigade of the Army of Northern Virginia where they became part of the famous Louisiana Tigers.
The companies in the regulars do not show a hometown of origin as do most volunteer companies. I think they were recruited to be a regular army unit to serve in both peace and war.