I started with street directories
http://www.lennonwylie.co.uk 1880 section on Donaghadee
no Fulton or M’Mahon or McMahon there is a John Fullerton publican Back st [I didn’t try
https://www.nidirect.gov.uk/information-and-services/search-archives-online/street-directories]
I tried Ros davies Co Down website –
http://www.rosdavies.com/SURNAMES/Mc/McMahon.htm has 16 Donaghadee entries
http://www.rosdavies.com/SURNAMES/F/FuFy.htm has 12 Donaghadee for Fulton
next their departure to South Africa
neither John Agnes Amelia Fulton or McMahonare listed in GenesReunited lists of migration. the best site being findmypast where i am not a member
next tried
http://www.ancestryireland.com trying for Agnes birth
AGNES MCMAHON 1845 FRANCIS Down Log in Register
AGNES MCMAHON 1851 JOHN Down Log in Register
AGNES MCMAHON 1852 JOHN Down Log in Register
AGNES MCMAHON 1856 WILLIAM Down Log in Register
AGNES MCMAHON 1860 JOHN Down Log in Register
AGNES MCMAHON 1861 PATRICK Antrim Log in Register
AGNES MCMAHON 1862 JOHN Down
http://www.ancestryireland.com for Agnes marriage
UHF has MCMAHON AGNES 1862 JOHN Antrim Log in Register
MCMAHON AGNES 1862 WILLIAM Antrim Log in Register
MCMAHON AGNES 1868 THOMAS Armagh Log in Register
MCMAHON AGNES 1879 JOHN Antrim Log in Register
MCMAHON AGNES 1886 JOHN Down Log in Register
MCMAHON AGNES 1893 WILLIAM Down Log in Register
MCMAHON AGNES 1903 JOHN Down
Some graveyard inscriptions exist see
https://www.ancestryireland.com/family-records/gravestone-inscriptions/?search1=amelia&search2=fulton&condition=begins&submit=SearchFULTON Donaghadee CI Donaghadee Donaghadee Down Log in Register
FULTON, MITCHELL, BROWN Millisle Pres Donaghadee Ballymacruise Down
FULTON Templepatrick Donaghadee Miller Hill Down Log in Register
FULTON, MODERATE Templepatrick Donaghadee Miller Hill Down Log in Register
FULTON, MODERATE Templepatrick Donaghadee Miller Hill Down Log in Register
FULTON Whitechurch Ballywalter Whitechurch Down Log in Register
FULTON Whitechurch Ballywalter Whitechurch Down Log in Register
FULTON Whitechurch Ballywalter Whitechurch Down
FULTON Whitechurch Ballywalter Whitechurch Down Log in Register
FULTON, BECK, STEWART Whitechurch Ballywalter Whitechurch Down Log in Register
FULTON, CAVAN Whitechurch Ballywalter Whitechurch Down Log in Register
FULTON, WRIGHT, McWHINNEY Whitechurch Ballywalter Whitechurch Down Log in Register
FULTON, YOUNG, DOGGART Whitechurch Ballywalter Whitechurch
McMAHON, CROTHERS Donaghadee CI Donaghadee Donaghadee
I tried census 1901 couldn’t find any McMahon in Donaghadee area same with Fultons although they were in nearby Ballywalter and Carrowdore
finally tried
https://www.irishgenealogy.ie/en/ where I found only a civil birth registration for Amelia Fulton at Newtownards, and presumably you have a copy?
so not much help, hoping others join in
Good luck