The parish of Drummaul exists all right. The small town of Randalstown is in the middle of it.
I assume your ancestor was RC. If so, the parish records don’t start till 1825, so you won’t find any record of someone who had moved to Glasgow by 1820. No easy way round that.
Drummaul is a rural agricultural area and there were never any city directories for a location like that in Ireland.
Research in Ireland in the 1700s is notoriously difficult due to the general lack of sources. Can’t think of any in Drummaul that would likely mention Bernard. The surname Murray is fairly common in that area which won’t help either.