Dicy/Dicey was a colonial Virginia nickname for Diana and was not used in New England, mainly because of its pagan connotations. From Virginia it spread with migrations. In the early 1800s as new styles of given names came along, Dicy was appropriated as a nickname for names like Laodicea, Candace, with a similar sound in them to the -cy in Dicy.
Dicy also became a name in its own right in the 1800s, like Betsy (Elizabeth) and Nancy (Anne/Agnes/Hannah) did.
Betsy was a common nickname for Elizabeth. Are you sure your associates were talking about Dicy and not Betsy? She could have been Elizabeth Diana or Diana Elizabeth. As a general rule, however, Dicy is not a standard nickname for Elizabeth. Of course, in the 1800s people were doing pretty much whatever they wanted to with names anyway.
The -cy/-sy pattern of nicknames, with the exception of Betsy, was more common in Virginia around 1800 than in New England. Betsy was common in both Virginia and New England.
Diana and Delilah are not names that go together particularly, even though they are both from the Bible. They are not in the same Bible stories. The rule of thumb is that if 2 names go together in the Bible, they go together in genealogy.