This looks like normal needle shed in autumn, which is common for arborvitae, pine, and several other evergreens. You can check out Ohio State's blog post about this as a comparison. Although we consider these types of plants evergreen, foliage does not live forever, so each year the oldest leaves/needles (which tend to be on lower limbs and towards the base of the branch on the interior of the plant) turn yellow or brown and then shed. It can be exacerbated by plant stress, yes, so both underwatering and oversaturated soil might contribute to more browning, but we do not see symptoms of root rot.

Usually, arborvitae dying of substantial root loss (in that case often from overwatering or poor drainage) get a sickly green color to their needles as they dry and become brittle, and they tend not to turn golden brown first; this color change will also include all young growth out to the branch tips and would not be limited to just the lower and innermost foliage. If anything, drought or under-watering prior to the heavy rains may have been the trigger here, or it's just due to the minor stresses of transplanting and establishment since they're relatively new. This does not look like an abnormal amount of shedding, so the plants will likely be fine. You can feel the soil around six inches deep to monitor when they'll need watering next; water only when it's becoming relatively dry to the touch at that depth.

Miri