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Does the Earth's orbit around the sun exhibit nodal precession, as does the moon's around the Earth? If it does, that must mean that the ecliptic is in a periodic wobble with respect to the celestial sphere?
The ecliptic is in periodic wobble wrt the celestial sphere, but that's because of nutation, not precession. Precession (nodal precession) changes where the Sun is during the equinoxes and solstices. The Spring Equinox, for example, has moved from Aries (it's still called "the first point of Aries") to near the Pisces-Aquarius border