Why does the moon orbit the earth? Shouldn't they both orbit a combined center of gravity?

You are correct: two objects orbit a barycenter. The moon's mass is less than 1/80 of the Earth's, so the barycenter is very close to the center of the Earth.

"The Moon is exceptionally large relative to Earth: a quarter its diameter and 1/81 its mass.[55] It is the largest moon in the Solar System relative to the size of its planet,[i] though Charon is larger relative to the dwarf planet Pluto, at 1/9 Pluto's mass.[j][115] Earth and the Moon are nevertheless still considered a planet–satellite system, rather than a double planet, because their barycentre, the common centre of mass, is located 1,700 km (1,100 mi) (about a quarter of Earth's radius) beneath Earth's surface.[116]"
en.wikipedia.org/wiki/Moon