Your Coull who died in Cullen and married an Aberdeen girl are unlikely to have originated from the Montrose area. There were a group of Coulls in Morayshire who had, as far as I can tell after many years of research, no connection with the Coulls of Angus.
Most Alexander Coulls from Moray came from either Rathven or Bellie, and if Alexander was illegitimate, I would expect the case to have been mentioned in the Kirk Session minutes for one of these two parishes. The birth might not have appeared in the baptismal register, as some ministers (just a few) refused to baptize illegitimate children.
The original kirk session minute books are in the NAS in Edinburgh, but copies may have been made, or the originals transferred to the archives dept. at Moray District. I am checking on that.