Washington/ Oregon Cascade Mountains

May is the most beautiful month in Western Washington and you will not be sorry. What kind of campgrounds are you interested in as there are many dry camps that are beautiful and will be open. Depending on the road conditions (and I think you will be fine in May) plan to come in on Hwy 2 instead of I 90 as there are a lot of National Forest campgrounds and towns Chelan just can't be missed. (Plan to take the boat ride up the lake. You can bypass going into Seattle by heading south on 203 until you get to I 90 and then taking Hwy 18 to 167 to 512 to 410 which will take you to the north side of Mt Rainier and the Sunrise Visitors Center. (Check as 410 will be closed if there is snow) Alternate would be to take Hwy 512 to Hwy 7 south and go to Paradise, which is the southwest side.
Mt St. Helens is interesting but don't plan to spend much time there as its just a hole. Head down to the Columbia River and drive up the river on Hwy 14 as there is tons to see and do along both sides. If you have time, drive as far as Hwy 197 and cross over there. But make sure you check out Bonneville Dam, and Multnomah Falls. I84 will take you to I 205 so you can bypass Portland and go to Mt Hood on Hwy 26

Beautiful places most tourist miss.
Baker Lake, camp spots are free after Labor Day until Memorial Day.
Fort Casey It's a state park right on the water (no hook ups)
Deception pass state park.
Newhalem, Seattle city light tour, alpine lake boat ride and dinner crew style.
Skaget Valley tulip festival, happens April usually.

I'm sure the locals will add to the list.

Late May, about the only roads that probably will still be closed are highway 410, the north Cascade highway 20. and the park roads leading to Paradise.
The roads into Crater Lake should be open.
all the major pass highways should be open. Even highway 20 may be open. (most years it opens about 1 June.)

Mount Rainier National Park - Ohanapecosh Campground and Cougar Rock Campground don't open till May 24th (Memorial Day Weekend and are walk up till late June).

Also keep in mind that you will need reservations just about anyplace you go. Our state parks are booked nealry solid 9 mos in advance. All the better sites are gone by now.

Check opening dates and don't go too high in elevation. There are so many places to go, it is hard to give you advice. Just wander around. I was in the woods several days a week year around. Expect rain every day, then you can be happy if you see the sun. Bring a tarp to put up over your fire.

Many wont be open that early. Depending on snow pack the rest of the season higher elevation camp geounds,could still have snow on the ground. Most of the campgrounds in the mountains are NF camp grounds. Services could be limited due to the gov shutdown. Expect cold and rain that time of year.