I had some contact a year or two ago with one of the Texas Fullagar's, any/all of whom until then I didn't know existed. [See story below.] I believe we established that the Texas folks were connected to the Penn Yann, NY, Fullagars. I believe the connection was late 19th century, but I may have that wrong.
Although my father's family was in New York state, the connection to the Penn Yann Fullagars is a bit unclear to me and I believe it was 18th century and in England.
My father, William Alfred Fullagar, was born 1915 in Western Spring, IL, but his parents (George Fullagar and Maude Humphries Fullagar) came there from NY shortly before his birth and returned shortly thereafter. Dad never knew why, but we have supposed that his father, a carpenter, expected work prospects to be better there and found it not to be so.
The tale of hearing from Texas Fullagars was one of these odd synchronistic happenings. We didn't know there was a whole bunch of Fullagars in Texas. Then one of my adult nephews, Scott Fullagar did a web search on his name. One of three things that popped up was an obituary that mentioned "Scott Fullagar" as a survivor of an elderly woman who had died in Texas. She had married a Fullagar, and later married another man. Scott (my nephew) was surprised. A day or two later, because of my visibility on the web, I heard out of the blue from a Texas cousin of the Texas (surviving) Scott Fullagar.
This is a long post to say I really can't answer your question,except to suggest looking toward Penn Yann.
By the way, I've encountered some suggestion along the way that Gallagher and Fullagar, along with others like Folger (of coffee fame) may in fact have started out as the same name. After all, spelling of names (and common English words) wasn't realy regularized until the 18th and even 19th century. So not only might I be a fifth or sixth (or whatever) cousin to your wife, but a seventh or eighth (or whatever) to you as well!