More detail about your family would help. The surnames are common in Ulster and Scotland, so more information is needed to narrow any searches. Give us the details of the ancestors who were born in Tyrone and rough dates of when they were born/married/died or left.
I wouldn’t worry too much about spelling. The idea of a single or correct spelling for a surname or a place name in Ireland is very much a recent phenomenon designed to meet the needs of modern officialdom. Before that there was no consistency. Names were spelled phonetically and each variation was down to the whim of the particular person recording the information. You will often see the spelling change as the records go back. This rarely indicates a deliberate decision to alter the name, nor even a mistake. Not everyone was literate, but even when they were, exact and consistent spelling simply wasn’t something they bothered about. And when folk from Ireland went to Scotland the spelling often changed to accommodate local customs, so for example, McDonnell often became MacDonald. Acheson/Aitcheson could be the same.