The name Morrow is fairly common in Donegal. The majority were Protestant suggesting they were incomers. (1901 census has 435. All Protestant save for 51 RC).
Co Donegal was settled by significant numbers of Scots and English in the 1600s, so a there’s a fair chance your family came over from Britain sometime in that century. There are no Morrows in Donegal in the 1630 Muster Rolls so that suggests they arrived after that.
MacLysaght’s “The Surnames of Ireland” says of Morrow: “Without the prefix Mac, Morrow is now more common in Ulster than in England whence it originated.” So it looks likely your family are of English origins.
We have no individual records of folk who settled in Ireland in the 1600s. We do know the details of the main English & Scottish landowners but not of the folk who settled on those lands.