My family is also of the dark haired (black or dark brown) ,brown or hazel -eyed variety of Irish (some with slightly sallow skin,other who were lighter and tended more to burn/freckle.) This look is found very strongly through the celtic areas, and also in many parts of England.
The first settlers in both countries came from ancient Iberia (Spain/Portugal) between 5-10,000 years ago. The celtic people aren't technically 'celts' at all (celts proper were Iron age people from central Europe), but they are culturally celtic.
Even England is not as 'Anglo-Saxon' as once thought; estimates vary depending on the researcher and criteria, but it could well be under 50% with the rest sharing similar dna to their celtic neighbours. In western England, this is definitely the case; the west has about 75% native British ancestry.
I had my dna tested,and it's likely my oldest maternal ancestor reached Ireland at approx 5000 years ago, by sea, following a trade route through the Mediterranean, up the coasts of Spain/France, and then obver to Ireland. They would have been early farmers and possibly megalith builders.
*the 'look' you describe used to be known in old-school anthropology, as Atlanto-Mediterranean, being found mainly on the Atlantic coasts of Europe from Spain on through the British Isles/Ireland