Dunbar to North Berwick is unusual because the only fares available are for advance tickets, routed XC & Connections. This will have arisen due to the clusters Cross Country have used for setting their advance fares.
Drem to Dunbar is a more unusual case in that both stations are served by Scotrail but the Dunbar services are non stop from Musselburgh, passing through Drem without stopping.
So other than via Musselburgh there's no service which to my mind is crazy
Walyford,Prestonpans and Longniddry are similarly involved
Interestingly an article has appeared in today's(15/8/19) East Lothiam Courier covering this very point!