The 38-year-old Woodard, from Greer, S.C., was the only player to finish under par.
"It was chilly starting out, but the big factor was the wind," Woodard said. "There was at least a two- to three-club difference from yesterday. I just tried to err on the safe side and take big numbers out of play."
Meghan Stasi of Oakland Park, Fla., finished three strokes back after a 71 in the event limited to players 25 and older.
Defending champion Ellen Port advanced with rounds of 76 and 75. The 51-year-old Port, from St. Louis, won the USGA Senior Women's Amateur last month.