Hello Laura,
I'm not sure this will be of any help, and I only mention it because of the Bellaghy connection, but in 1794 two Downing brothers, the sons of Dawson Downing of Bellaghy and his 1st wife Catherine Fullerton, changed their name to Fullerton in order to inherit from Alexander Fullerton of Ballintoy the uncle of Catherine Downing nee Fullerton who by 1794 had died.

The eldest son was named George Alexander. The youngest was David Fullerton so at the name change he became David Fullerton Fullerton.

George Alexander Fullerton's family has been traced to the present day but nothing is known about David Fullerton Fullerton so much so the present day relations of George had no knowledge of his brother David. But he must have married and had a child as I have a picture with an inscription on the back -

Lieut. J.C. Fullerton R.N. painted by Mrs J.C. Fullerton 10th September 1841.

The picture is difficult to age, but he is round faced with long red sideburns and hair which recedes from the sides up onto the top of his head showing a very wide forehead.

The above year dates are the only two I have. On 26th May 2008 8:27p.m. I posted a longer email entitled
'Fullarton/Fullerton DNA Family Study - Name change from Downing to Fullerton'. Explaining how a Fullerton used to be a Downing.

I hope some of the above helps. Good luck. Jill