Hawaii is really nice, but not sure how dry since it's a tropical climate... with the wind blowing might be okay... Honolulu is a big city with lots of vehicle traffic I'm sure... haven't been back there since the mid '70's when I worked for an airlines and could fly there pretty cheaply. The big island would likely be the least congested, but has more active volcanoes...
They have a dry side and a wet side of the islands, the dry side is pretty good - if the house is kept open/aired out. The wet side not so much, mold everywhere. I'd ~mostly~ avoid the Big Island for the most part the volcano has periods where the sulphur compounds combine with the fog to produce "vog" (volcanic fog I guess). Its nasty stuff, not there all the time but icky when it is. The intensity varies widely depending on exact location on the island though, some of the north end and especially north end of the dry side isn't so bad. We'd get a whiff on Maui occasionally but not enough to matter. The main problem we'd have on Maui was when they'd burn the sugar cane, but the last sugar plant there is shutting down so I'm thinking that's a non problem anymore. If you really like rural, Kauai is really nice.. but is indeed pretty rural.
I think Oregon is a beautiful state and when I visited the kids in Portland, told my wife that I could live there... I liked Portland and the surrounding area, but as Ryan pointed out, lots of greenery that could cause problems...
Actually its sort of a trade off
The wet side (west) plants tend to be not as bad for allergies in general but you get more mold and moss. The dry side (east) plants tend to have brief glorious lives with a TON of pollen. There are a couple of exceptions of course - in parts of the Willamette (kind of wet side) they raise grass seed and that's a big NOPE on our list. We're kind of splitting the difference here, we're in scrub oak country so wet enough that some trees can still grow, but dry enough that most of the moss/mould problems are only a minor nuisance in the winter months.
Further out on the dry side its REALLY dry -
https://en.wikipedia.org/wiki/Rattlesnake_Mountain,_Benton_County,_Washington "Rattlesnake Mountain is often described as the tallest treeless mountain in the world" - probably not actually true, but there are no trees growing there because its to dry.
As far as really nice climate in Oregon the hills above Medford (not down in Medford itself - its another smog sink) either north towards Roseburg or south towards Asheland is really quite nice. Asheland is a bit higher up so a smidge cooler/clearer air and also has a pretty decent art scene. The prices go up substantially when you get close to there though.
Generally in most of the west the rule of thumb is "look for a drain downslope" where the bad air can escape, if you don't see it you're in it and move on.
The dry side is effectively the same as a lot of the desert west, the exact vegetation varies slightly but its almost indistinguishable from high desert in Arizona and low river valley desert in BC if you squint just a little. The low desert areas in the southern states are a different ball of wax of course.