I have found another William. On the 1910 Washington Parish census there is a William Mullina [double "l"]. He is 45, married 22 years with 9 children. His wife is Fannie and his oldest child is the Loyd H. who was the father of the first William that I found. Fannie died 7/1/1924 and was a widow at the time. Her father is shown as James Lowery.
In 1900, William is shown as William Malina. According to the census, he was born in LA, his father was born in NJ and his mother was born in LA. In 1880, I found him living with Thomas M. Stevens [33, MO, CT, SC] and his wife Tabitha M. [25, LA, LA, LA]. His name is William H. Malany, but was picked up as Malony. He is 19 and is shown as a nephew. I presume he is nephew to Thomas. On this census, his father is shown as being born in NY. This couple is shown as living next door to William in 1900 and 1910.
I have not yet found William on the 1870 census. It is probably because his name is spelled differently.
On the 1900 census Loyd H is shown as Hardy L. So, the H in William's name could be Hardy rather than Henry.
I'll see what else I can find.