The Weaver Creek Public Fishing Area is a great place to launch paddle-powered craft, and also includes a small cleared area for bank fishing. Some folks also fish right off the ramp, especially after dark, for cats.
North Carolina Wildlife Resources Commission and search for "Weaver Creek" in the Chatham County section.
As you can see from their picture, you can't back a trailer down that ramp. I don't know what the rules are -- it's often said "no motors", but there are occasional boats out there with what seem like 5hp jobbers.