I can't think of any street in Wilmington or northern Delaware where the suffix is dropped and people still know they are talking about the street instead of the place (or person or anything else). The closest we have to that is Delaware Avenue in Wilmington, which some people say as "Delaware Av". But it doesn't even really work that way for Pennsylvania Avenue, which kind of parallels Delaware Avenue.
In Delaware, it seems easiest to call most streets, roads, highways, and expressways by some name with three or four syllables. Two syllables seems kind of unusual around herel; the suffix or prefix is deliberately added to get to three syllables, if it would otherwise be two syllables if you only used the identifier. Most significant streets and roads either have 3 or 4 syllables when talking about them: Union Street (in Wilmington), Delaware Av (Delaware Avenue in Wilmington), Limestone Road (suburbs), Kirkwood Highway (suburbs), Concord Pike (suburbs), Naamans Road (suburbs), Route Thirteen (US 13), One Thirteen (US 113), Two Oh Two (US 202, also called Concord Pike north of Wilmington), I Ninety Five (I-95), Two Ninety Five (I-295), Four Ninety Five (I-495), One Forty One (DE 141), and so on.
EDIT: I can think of one street where the suffix isn't used all the time. New Linden Hill Road in the Pike Creek area outside Wilmington is sometimes called "New Linden Hill". But there is no place called New Linden Hill like the original post, nor is there a Linden Hill.
On the other hand, if you are a true Delawarean, you know what street someone is talking about when they say they are going to "the Avenue".