I have never come across this story before and, I suspect, for good reason. It could only have been done with vast expenditure of effort, at a time when when there was a dearth of manpower available. The effluvia from the decomposing bodies would have seeped out of the lower end of the cut and contaminated the waters of one of the most important ports in the kingdom. I would further point out that Winchelsea is the proud possessor of a large, circular, tree covered mound in the open fields on the south end of town close to the New Gate. Of course, there is no proof that it was the burying place for plague victims but it is clearly man-made, and is at least a credible location.
Folk memory does tend to be both long and accurate.
My understanding from reading the literature on Winchelsea is that the name Deadman's Lane was coined after one of the great French raids on the town in the 14th century. At that time, St Gile's Church stood above this deep track down to the shoreline. They were found there after the French had vacated the sacked and burned town and survivors crept back to face the devastation. The current explanation really makes no sense and is at odds with both published commentary and local knowledge. In fact I was woken by the fact that the air in my bedroom was becoming unbreathable because I had my window open.