Comparing County destroyer displacement

Were the County-class guided-missile destroyers much smaller than the later destroyer being compared?

No. The figures given are relatively close, although standard and full-load displacement differ.

The County class is given a displacement of about 6,200 tons. The later design is quoted at about 6,300 tons standard and 7,100 tons fully loaded. Those figures do not support a large difference in size. A different hull design does not by itself establish a much greater displacement.