I found this list on tidalflats.com. I think the options are pretty limited for beach access during the spring and summer.
State and National Parks: Dogs allowed on beaches, year round, on a 6-ft. lead. Dogs prohibited from bathing areas when guarded at Island Beach State Park and prohibited from Ocean Beaches at Sandy Hook National Seashore from March 15th through Labor Day. There is a similar restriction at Townsends Inlet State Park and Cape May State Park (April 1 to Sept. 15).
Monmouth County Beaches: dog beach on Manasquan Inlet (Fisherman's Cove) open year round. Dogs should be on lead.
Ocean County Beaches: Mantoloking allows dogs on lead year round but are restricted between 8 am and 6 pm from May15 to Oct 1. South Seaside allows dogs on or off lead in the off season (Sept 15 to May 15). Seaside Park does not allow dogs on beaches, year round.
Long Beach Island: Beach rules vary by township but (where allowed) dog use is generally excluded from May 1 through Oct. 1. However, we found Beach Haven to be very dog friendly through Mid-May. Prior to Memorial Day and after Sept. 30 dogs are allowed on the beach, off-lead. There is a year-round dog beach on the bay side in Beach Haven at the end of Taylor Ave. The is also a bay beach at the end of Joan drive in Long Beach township that allows dogs from Oct. 1 through Mid-May.
Brigantine Island Brigantine has a dog beach on the northern end of the island that allows dogs year round, on lead
Atlantic County Ocean City allows dogs from Oct. 1 to April 30.
Cape May County: Avalon/Stone Harbor allow dogs anytime between Oct. 1 and Feb. 28. Otherwise restricted. North Wildwood allows dogs from Sept. 16 to May 14. Cape May Beaches allow dogs only from November through February.
Higbee Beach WMA: dogs allowed year round on leash.