Schultz, the 2006 runner-up at this event, scored a 1-up victory over the 2000 Women's Mid-Amateur titlist Moore, who was trying to become just the second medalist to win the title. Carol Semple Thompson did it in 2002.
Schultz next faces Madelaine Campbell, a 2-and-1 winner over Terry Mayes.
Semple Thompson, playing in her 106th USGA championship, lost Tuesday to Lisa Smego, 1-up. Smego next faces Robyn Puckett, who beat Toni Wiesner, 3-and-2.
Two-time defending champion Diane Lang was also a winner in the third round, defeating 2004 titlist Carolyn Creekmore 1-up.
Lang, attempting to become just the sixth three-time winner of this event, next faces Noreen Mohler, who beat Natalie Easterly, 1-up.
There will be two rounds of match play on Wednesday. The 18-hole final is scheduled for Thursday.