Alison,
I see David Alexander listed in Lisnagunogue in the 1803 agricultural census of Co Antrim, on Bill McAfee’s website:
http://billmacafee.com
He is also listed in the townland in the tithe applotment records for 1824:
http://www.irishgenealogyhub.com/antrim/tithe-applotments/billy-parish.php
He is gone by 1861 and so presumably had died before that.
I looked for a death for Martha Alexander in 1878 around Ballymoney. I did not find one for that year. Nearest was a Martha who died in 1876 aged 91. She was a spinster living in Islandmore. So therefore she doesn’t appear to have been David’s wife.
https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_returns/deaths_1876/020598/7227319.pdf
Griffiths Valuation for 1861 lists Joseph Alexander in Lisnagunogue Upper.
He had plot 4 which was a 14 acre farm. He died 5th Feb 1887 aged 81. The informant was his son Matthew:
https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_returns/deaths_1887/06227/4781911.pdf
Joseph’s will is on the PRONI wills site. It mentions his wife Betty, daughter Martha & his son Matthew.
Joseph’s wife Elizabeth (Betty) died in 1899. Informant was again her son Matthew:
https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_returns/deaths_1899/05799/4640706.pdf
Joseph & Elizabeth were married in 1852:
https://civilrecords.irishgenealogy.ie/churchrecords/images/marriage_returns/marriages_1852/09427/5415005.pdf
Here’s the birth of their son Joseph in 1864:
https://civilrecords.irishgenealogy.ie/churchrecords/images/birth_returns/births_1864/03619/2335662.pdf
Death of daughter Martha in 1891 aged 32:
https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_retu…
This looks to be their son Matthew in 1901, farming nearby in Feigh:
http://www.census.nationalarchives.ie/pages/1901/Antrim/Dunseverick/Feigh_or_Dunseverick/923060/
Here’s their son Matthew’s marriage to Martha Mitchell in 1887:
https://civilrecords.irishgenealogy.ie/churchrecords/images/marriage_returns/marriages_1887/10807/5941462.pdf
Same family in 1911 (13 children, 11 alive):
http://www.census.nationalarchives.ie/pages/1911/Antrim/Dunseverick/Feigh__alias_Dunseverick/114327/
Probate abstracts:
Alexander Matthew of Feigh county Antrim farmer died 11 January 1923 Probate Belfast 29 October to Martha Alexander widow Robert John Alexander and Matthew Alexander farmers. Effects £941 15s.
Alexander Robert John of Feigh county Antrim farmer died 28 December 1924 Administration Belfast 20 December to Joseph Alexander farmer. Effects £10.