I think I'd call them Cupressus rather than Thuja. They are known to have issues with canker in the east, I'm sure someone in your neck of the woods will chime in.

Those look like Cupressus macrocarpa 'Wilma' or 'Wilma Goldcrest' (sometimes just 'Goldcrest'), not thujas. Rub your hand through the foliage - if that releases a distinctly citrus aroma, then that's what they are (also known as lemon cypress). They need very well drained soil - more so than thujas - and can be damaged by cold. In our climate, I'd be concerned first about soil conditions - much less with cold. Winter wet soils here are often the cause of plant failures that many attribute to cold damage.

They tend to be rather problem free otherwise, Coryneum canker can be a problem but not in the PNW - that's more of an arid climate issue.

The soil, at least for the first inch or so, doesn't seem overly wet. How do I tell what is appropriate? If it is too wet, what can I do?

'Wilma' is a smaller growing, less yellow sport of 'Goldcrest'. Many others in the area have brown patches this spring, just like those on yours. Whatever the specific cause timing would suggest it is something that develops during winter. As you can now see - with them growing above gutter height - these want to grow bigger than the location you have them in allows. And in addition to the browning these tend to fall open, at least in this region. Although it has by this time been highly abundant on the local market for some years, I still see few lasting, successful examples. A.L. Jacobson, Trees of Seattle - Second Edition (2006), who deems them "Likely to be killed or hurt in severe cold" nevertheless does also mention a 'Wilma'/'Goldcrest' 6 ft. tall on Federal Ave E, and another approaching 8 ft. tall near E Blaine St. Of course, such heights are peanuts for this species, for instance 48th Ave S had a normal (wild type) green one about 75 ft. tall during the same period.

Would transplanting them be an option? With the location they are in now, I wouldn't be supprised if the answer is no, but it's worth asking.

Too late in growing season for transplanting. Try cutting out dead parts and shearing the rest back for now. Late spring-early summer usual recommended timing for shearing of conifers. Stay out of bare interior with maintenance shearing of these (and most other conifers) or you may be looking at bald sections indefinitely.

I'll try that. Thank you for the advice.