Coughlan with an 'a' is certainly the standard spelling in Co. Cork. It is unlikely that your Coughlins actually 'changed' the spelling of the name if they emigrated before, say, 1900. Officials, or priests, usually wrote down what they thought they heard, or what the thought they had previously seen, whether or not the individual could read or write. The name 'Timothy' was no more than a conventional spelling for the Irish name Tadhg and the spelling was taken from the biblical Timotheus early on, though it is doubtful that anyone was actually called Timothy until quite late. The parish priest would hear Tadg (or variants like 'Thady") and write down 'Timothy'. Such a spelling eventually influenced the pronunciation of the name, and in the U.S. priests generally wrote 'Timothy' no matter what they heard. It would probably be the most practical solution for you just to write Coughlan and Timothy.
