I think I have found out a little about your "family secret". I found Odile Blanchard on the 1860 census for St. Martin Parish, Grande Pointe. She was 24 and living with her widowed mother and numerous other family members including a sister, Josephine. The faimly lived at house #408. The page # is 52.
In 1870, Odile Blanchard was the head of household in Ward 1, St. Martin Parish. Census is on page 442. Living with her were Albert Tally [3 months old], Joseph Savoie [10 years old] and Edward Jennings [37 years old]. Joseph and Edward are shown as swampers. I think that this shows that the two boys did not have the same father. I went back to the 1860 census and found that in house #406 there lived a W.D. Tally who was 35 and a planter. He lived alone. However, there was a woman and child with that surname living in Grande Pointe. I suspect he may have been the father of Albert.
In 1880, Joseph Blanchard [no longer using the Savoie name], 21 and Albert Tally, 10, were living with their Aunt Josephine and her family. She was married to Aristide Cormier. Both boys were farm laborers.
The next census is 1900 and I did not find Albert. I did find Joseph in 1900 and 1910. If you live in Louisiana, you should be able to find the birth/baptismal records for the two boys. They would be in Father Hebert's books. Joseph would have been born about 1860 and Albert about 1870. You could try Blanchard first and if you don't find them, check Savoie and Tally. Sometimes the spelling of Tally looked like Tully, but I think it was Tally. Albert may have died after the 1880 census. You could check for a death record for him between 1880 and 1900. I forgot to check for a death record for Joseph. It should give his father's name. If I find one, I'll let you know.
Hope this helps. If you have any questions, write and I'll try to answer.