I am volunteer honcho of the Alexander DNA Project. As such I collect the Alexander lines of our members. I can tell you we have one member in our project who descends from your Archibald. Our member describes your Archibald as follows:
"Archibald Alexander (Sr.) b. 1708, Manor Cunningham, Ire. , m. Margaret Parks, 1734 Cty.Donegal, Ire., m. (margaret?) Jane McClure after MPA d. in Augusta Cty., Va. in 1753. AA (Sr.) d. ca. 1780, Augusta Cty., Va."
The son of this Archibald whom our member descends from is listed as:
"Archibald Alexander (Jr.), b. 1771 Augusta (now Rockbridge) Cty., Va., m. Isabella Patton, 1795, Augusta Cty., Va., d. ca. 1830, Bracken Cty., Ky."
This member does NOT list a father for your Archibald Sr.
For your information, this member has 2 close Alexander DNA matches whose earliest TWO generations are:
"Robert Alexander b 1720 Donegal IR, d 1783 VA, m Elizabeth Beard b 1725 Augusta Co., VA" through his son "Thomas Alexander b 1746 VA d 1808 Letart, Meigs, OH, m Elizabeth Buchner"
"Thomas Alexander b. 1803 Charleston, VA, m. Delilah Sailor 18 feb 1823 Rush Co., IN, d 1874 Wabash Co., IN. bro James Alexander NC" through his son "Reuben Alexander b. 1823 Rush Co, IN, d 1902 Crawford Co., OH; m. Hanah Churchill b. abt 1831 OH"
All of this is shown on our website at http://www.familytreedna.com/public/alexander-y-dna/default....
If you want some contact information or anything else, I am at firstname.lastname@example.org
Personally I am not related to this Alexander family.
I hope this helps.