Some exhibits have live animals, some of which are seldom seen by regular folks. Just across Bicentennial Plaza from Natural Sciences is the N.C. Museum of History. It has more than 55,000 feet of exhibit space and a 315-seat auditorium. Right down the street from the museums is the N.C. State Capitol, as fine an example of Greek Revival architecture as you'll find anywhere in the state. The building was constructed from 1833 through 1840 and is now a National Historic Landmark.
Tours are available.
Head a couple of blocks east and a couple north, you can visit the N.C. Executive Mansion. Tours are available and admission is free. If you head south from the Capitol, look for the Raleigh City Museum in the Historic Briggs building at 220 Fayetteville Street Mall. Admission is free though donations are encouraged. With all the walking you've done by now, you should be.