I checked Father Hebert's Southwest LA records and found records of marriages of some of Charles Harrington and Sally Ashley's children, but that was it.This family was not Catholic, so only the marriages were recorded when the children married Catholics.
Children of Charles and Sally: HARRINGTON, Charles - of this parish; native from Mobile (minor son of Charles - inhabitant at la petite anse & Celeste HASEBEY)m. 19 Aug. 1817 Eugenie DEROUEN - native of this parish (minor daughter of Joseph DROUAN - inhabitant at la petite anse & Marie PREJEAN)Wits: Joseph Fortune PENNE, Louis LINGOIS, Louis DUSOUCHET, Billy HARRINGTON, Josephine DROUAN, Bethsy HARRINGTON. Fr. Gabriel ISABEY (SM Ch.: v.6, #54)
HARRINGTON, Rebecca - a Protestant, native of Kentucky (minor daughter of Charles - of la petite anse & Sarah HARRINGTON) m. 4 Oct. 1815 James NEWMAN - a Protestant, of this parish and native of Pennsylvania (major son of Hyla & Nancy NEWMAN)Wits: William GREIG, Louis CHEMIN, Conrad ROSSI, W. GREIG, Lipton PURCELL, Thomas HARRINGTON.Fr. Gabriel ISABEY (SM Ch.: Folio F)
HARRINGTON, Elizabeth - a Protestant, of this parish and native of Augusta County, South Carolina (minor daughter of Charles - of la petite anse & Sally HASHLEY)m. 22 Feb. 1819 James HAYES - a Protestant, of this parish and a native of Barcley County, Virginia (major son of Gabriel & Mary SMALL)Wits: James NEMMAN, William GREIG, Conrad ROHN, William HARRINGTON.Fr. Gabriel ISABEY (SM Ch.: Folio F)
HARRINGTON, Polly - non-Catholic, of St. Martin Parish (minor daughter of Charles & Sally ASHLEY)m. 31 July 1823 at parish rectory toWilliam WALKER - non-Catholic, of St. Mary Parish (major son of Samuel & Jenny DAVIS)Wits: Thomas HARRINGTON,William ARMSTRONG, Charles HARRINGTON, C. H. LANDRY.Fr. Marcel BORELLA (SM Ch.: Folio F)
HARRINGTON, William - of this parish; native of the river [Mississippi] (major son of Charles - inhabitant at la petite anse & Sally HECHLEY [probably ASHLEY])m. 21 May 1822Sally FARK - native of this parish (minor daughter of George - inhabitant at Vermillion & Dalila HARTGRAVE [HARGRAVE] alias AlgrotWits: Solomon MORGAN, Emile ROUSSILLON, Horatio DAVID, Louis DUSOUCHET. Fr. Gabriel ISABEY (SM Ch.: v.6, #296)
*Note that a minor child is under 21;a major child is 21 or older.
I don't see a Charles Harrington of the appropriate age on the 1830 census, although it is possible he was living with one of his children and thus not enumerated.His sons appear to be living in Lafayette Parish.
I cheked the 1850 census as well -- the younger Harringtons are living in Vermillion Parish;there is no older Charles Harrington.The census does list Charles Harrington, b AL 1797 and William Harrington, b KY 1796, with the wives listed above.
I would recommend that you check the St. Martin Parish (and possibly Lafayette Parish) courthouse to see if they have a probate record for Charles between 1823 and 1830.It is likely that he owned land that would have needed to be passed down.There is a small possibility that the probate record could have mentioned any nieces or nephews or siblings that he might have had, to help you find his family.At the very least, it will help you narrow down his age.
You may also be able to find land records in KY in 1796 or 1797, since it is likely that that is the time when the family moved to the Mobile area.Tax rolls will also help you track his movements around the country.
If you can find baptism records for the children in the various places they claim to be from (SC, KY, and Mobile), they should list the names of the children's grandparents.