I moved three arborvitaes (1 five feet and 2 3 feet) from the front of the house to the side, which receives 4-6 hours of sun per day. Before transplanting I cut the dead branches of the small shrubs, which was a substantial amount. The soil on the side is clay and rock, so I amended the soil around the shrub balls with fertilized planting mix and peat moss. The area also slopes down and water is plenty, so I watered only the day I planted and the next day. It's been a week. One of the small shrubs has a whole section light green. Any thoughts why and if I can do anything to save it? Should I cut that part of the shrub to let all nutrients go the healthy-looking (so far) parts?

Sounds like the light green portion has already decided to die. Most likely the needles will continue to decline, shrivel and perhaps but not always change to brown. scraping the bark should reveal and dark green moist looking cambium. You can compare this to a healthy branch for comparison. If a dry light green, the plant has decided to let that section go and you could prune it away.