Son A appears to be:
.  Thomas CARRICO, b. 1800-1810; listed next to Margaret (CARRICO) REEVES in the 1830 census

1a.  Jordan R. Dodd, ed.  1993.  Kentucky Marriages:  Early to 1850.  Liahona Research, Orem, UT (Broderbund CD-229):

Callico, Josiah

Anderson, Catharine

3 Nov 1796

Jefferson Co.

Probably either the bond date or the license date, not the date of the marriage ceremony.

1b.  WorldConnect / Ancestry World Trees (online at RootsWeb.com/Ancestry.com).  A file there gives this abstract of their marriage bond:

Cairco, Josias (sig) Callico, Josiah (in bond); bd. Thos. Anderson, Jr. Anderson, Catherine; f. Thomas Anderson, Sr.  B: 11-5-1796 M: 11-3-1796

The bond date can't be after the marriage date, so surely the dates have been switched (the original needs to be consulted).  Some secondary sources say they were married on Nov 10th.  If so, presumably the bond date is the 3rd and the license date is the 5th (please see this page for a discussion of early marriage records).  We do find out that Catherine's father is Thomas ANDERSON, Sr., and that Thomas ANDERSON, Jr., is most likely her brother.

2.  1800 Census Index/Images (online at Ancestry.com):  the 1800 KY census was lost.

3.  Census Index (online at GenealogyLibrary.com):

1807 Voter List (Males over 21 Years)

Carrico, Josiah

IN

Knox Co.

p. 5

4.  1810 Census Index/Images (online at Ancestry.com):  there was no 1810 census in the Northwest Territory (Indiana did not become a state until 1819).

5.  1820 Census Index/Images (online at Ancestry.com, Image #10 of 14; extracted by Diana Gale Matthiesen):¤•

1820

IN

Sullivan [now Greene] Co.

Roll M33_15

p. 122-123

Ln. 16

Josiah Carrico

010001-31110-0-?00

These data indicate:

No. & Sex	Age Class	Therefore Born	Individuals Inferred
1 male	10-15	1804-1810	= Son A
1 male	45 or over	in or bef. 1775	= Josiah (b. 1769)
3 females	9 or under	1810-1820	= Daughter E = Daughter D = Daughter C
1 female	10-15	1804-1810	= Daughter B
1 female	16-25	1794-1804	= Daughter A
1 female	26-44	1775-1794	= Catherine
?	person engaged in agriculture

Daughter Margaret is married and living with her husband in Knox Co., IN.

6.  1830 Census Index/Images (online at Ancestry.com):  not found.

7.  Jack Baber.  1875.  The Early History of Greene County, Indiana…  N.B. Milleson, Worthing Times Office, Worthington, IN  (online at the Greene County INGenNut site):

p. 79

Chapter XXX Stafford Township Among the first old settlers in Stafford township, were JOSIAH CARRICO and JOSEPH DIXON.  Afterwards came BARTLETT GOODMAN, old MASON PITTS, ISAAC HUBBELL and JAMES WALKER.  The Carrico marsh was named for the old man Carrico, as that swamp was his pasture, where he kept his stock.  The first log cabin was built by the oldest settler, Uncle JOSIAH CARRICO, on the farm where JAMES BURRIS now lives... Creeks and Branches: Delaware fork of Bogard's Creek, in the west part of the township, was called so far the last tribe of the Delaware Indians that was seen hunting there.  That creek empties into the cane drain and passes out into Knox County.  The Goose Pond is a very large march and prairie, covering about five miles of soil with water and geese.  We have wild geese here every year by the thousands.  Carrico marsh was the old man Carrico's stock pasture and range for over twenty-five years. It has been a great place for old pioneer hunters.  Mr. CARRICO raised the first crop of corn in this township...

p. 80

... The Carrico log cabin was on the place where JAMES BUESIS now lives.

8.  Anon.  1884.  History of Greene and Sullivan Counties, State of Indiana...  Goodspeed Bros., Chicago, IL (online at — mea culpa, I forgot to note where I got this, not that it really matters):

HISTORY OF GREENE AND SULLIVAN COUNTIES, STATE OF INDIANA, FROM THE EARLIEST TIME TO THE PRESENT;  TOGETHER WITH INTERESTING BIOGRAPHICAL SKETCHES, REMINISCENCES, NOTES, ETC., CHICAGO:  GOODSPEED BROS. & CO., PUBLISHERS.  1884. pp. 11-12, 21, 288-295: Greene County was home to Native Americans and French fur trappers until the early 1800’s, when the first white settlers began to appear.  Goose Pond and Bee Hunter Marsh are in what became Stafford, Stockton, Grant, and Washington Townships in 1821.  The first settler in Stafford Township was Josiah Carrico from Kentucky, around 1817.  The following year, other settlers appeared and began clearing portions of the upland areas for crops.  The lowland areas were too wet to farm but were home to an abundance of wildlife, including waterfowl, wild turkey, deer, wolves, panthers, and bears, which provided good sport as well as a key food source.

8.  WorldConnect / Ancestry World Trees (online at RootsWeb.com/Ancestry.com).
A file there gives this quote in the Notes about our subject:

From History of Greene Co., Indiana , p.290 - "It is stated that the first settler in the present Stafford Township was a man named Josiah Carrico, or as he became universally known and is called to this day, Calico.  He was a long-limbed Kentuckian who had first settled in Sullivan County, probably in Jefferson Township, but had come to what is now Stafford Township as early as 1817, and possibly a year before.  He located in the southern part of the township on the border of what is yet known as the 'Calico Mash' (sic).  This was a large marshy, open tract of land which Carrico used as a pasture for his stock, and in which he is said to have killed many deer and a few bears.  He lived in the township until his death, having several children, one of whom it is said now lives in Terre Haute."