Yes, wax worms turn into moths. Waxworms are the caterpillar larvae of wax moths, and the two species bred commercially become the greater wax moth (Galleria mellonella) and the lesser wax moth (Achroia grisella).
The greater wax moth goes through four life stages: egg, larva, pupa and adult. A larva can stay in the larval stage for anywhere from 28 days to six months, molting eight to ten times, and then spins a silk cocoon. The pupa does not feed and stays in its cocoon for one to nine weeks before a brown-gray adult moth, 10 to 18 mm long, emerges. Adult females live about 12 days on average and males about 21 days. In the wild, wax moth larvae live in beehives and chew through beeswax, which is why beekeepers treat them as pests. Waxworms sold as fishing bait or reptile food can survive for weeks at low temperatures, and keeping them cool lets them go a long time without eating.
Source: Wikipedia, Waxworm
Image: Greater wax moth, Ben Sale, CC BY 4.0, via Wikimedia Commons.