Yes, both the Ó and Mac parts of a surname were often dropped when names were being "anglicized". The Gaelic word Mac, if it wasn't dropped, was anglicized in various ways, such as Mac, Mc, and even M'. So the Irish surname Mac Domhnaill, which means “son/descendant of Donal”, could end up as MacDonnell, McDonnell, or even M'Donnell, as well as just Donnell, and in some cases an "n" or an "l" was also lost. The surname is spelled MacDhòmhnaill in Scottish Gaelic (with the same meaning). The surname also sometimes became MacDonald. The “d” at the end of MacDonald is a later addition in the anglicized version, since (as shown above) there is no "d" there in either the Irish or Scottish original forms. MacDonald is more common in Scotland (and among Scots settlers in Ireland, especially in Ulster), but McDonnell and McDonald were both used in Ireland as anglicized forms of the surname, sometimes within the same family in various records (as happened with my own McDonnell ancestors).
Most of the Irish McDonnell's are actually descended from Scottish settlers (part of Clan [Mac] Donald) who came to Ireland to settle in the Glens of Antrim in the 13th century, from where, by the 15th century, they controlled all of North Antrim. If you've been to Ireland, the ruined Dunluce Castle which is a popular tourist site near the Giants Causeway on the North Antrim Coast was the main McDonnell family castle. A large number of those McDonnell's were brought from Antrim to what is now Mayo in the late 14th century, to assist the ruling family of that area, the Anglo-Norman Burke family, and my McDonnell relative are descended from that group.
I've mentioned that surname in particular, because it was also sometimes anglicized (in Ireland and Scotland) as MacDaniell/MacDaniel or McDaniell/McDaniel, and sometimes with an "s" added at the end, as in McDaniels. If the Mac/Mc part was then dropped, it could also become just Daniels. Most people in Ireland who use the McDaniel form are, however, descended from later Scots settlers (17th century or later), who came as settlers after the Reformation. While the earlier McDonnell settlers usually remained Catholic, these later distant relatives were already Protestant when they arrived.