I guess it depends upon when you want to stay. If you're talking about sometime between now and memorial day you'll have a few choices. After that, things can get a bit tight. If you're coming in the summer, book ahead.
It sounds as though you want to be not too far from a beach but have the option to go into the forest as well.
You could stay on the mainland in the vacinity of Island Beach State Park and drive west to Brenden T byrne State Forest, Wharton State Forest, Penn State Forest.
North of Cape May might be nice. You'd have beaches, part of the Deleware bay shore, great birding, Belleplain State Forest.
Look at an online map of an area where you think that you might like to stay and do a google or other search for 'motels'. This one was for Woodbine.
http://www.google.com/local?hl=en&lr=&ie=UTF-8&q=motels&near=Woodbine,+NJ+08270&sa=X&oi=localr
You might consider doing a search for southern nj motels dogs and see what pops up.
Sorry I can't recommend something specific, I don't have a dog and I've never stayed in a motel around here. I just live here.
Gillian