Best bets right now are the Haw and the Neuse (although the Haw is getting low).
Potential Haw access points are to put in at Chicken Bridge and take out at 15-501 above the dam (stay left of any islands to make sure you take out before this dam), and putting in below the dam and taking out at 64 bridge ( do not get close to the dam/backwash). Only paddle these sections below 1000 cfs (with a lower limit being around 400 cfs), as they get out of control at higher water.
USGS Real-Time Water Data for USGS 02096960 HAW RIVER NEAR BYNUM, NC
The neuse is runnable down to around 400 cfs, and there is a paddle shop that has canoe rentals and shuttles available right at the putin (paddle creek). Inquire with them on highest recommended level (right now it is 2500 cfs, which may or may not be too high). Best two sections are from Falls dam to Buffalo Rd (long and pretty), and from Millburnie Dam to Poole Rd.
City of Raleigh | Neuse River Canoe Trail (http://www.raleigh-nc.org/portal/server.pt/gateway/PTARGS_0_0_306_209_0_43/http;/pt03/DIG_Web_Content/category/Leisure/Parks_and_Facilities/Cat-2CA-2005920-142644-Neuse_River_Access.html - broken link)
Eno is nice, but it will only run with flash floods this time of year. Put ins/take outs are 1) Cades Fews Ford in Eno River St Park, 2) Pleasant Green, 3) Cole Mill Rd/Eno State Park 4) Guess Rd and 5) West Point on the Eno (Roxborro Rd...note you take out above a big dam, don't blunder into it). You will want a level of 300-900 cfs to run the eno.
USGS Real-Time Water Data for USGS 02085070 ENO RIVER NEAR DURHAM, NC
A note for the eno and Haw. Never put on when the river is rising after a rain, watch for it to drop into the desired range before you go, to ensure you don't get caught in a flood.
Enjoy