I doubt that the dips were ever rubbish dumps as they weren't in living memory and they wouldn't have been when New Brighton was at its prime as a major holiday resort, which started long before the sea wall was built.
I guess they were created when the sea wall was built (or shortly after) as flood water collection areas and that the term "infilled" has been interpreted as infilled with rubbish.
Alternatively they may have just areas kept low to shield them from sea breezes and were just for public amenity.
I may get proved wrong.
In a time of universal deceit - telling the truth is a revolutionary act. George Orwell
When the debate is lost, slander becomes the tool of the loser. Socrates