Washington state hwy 20 from Okanagan to Mt Vernon

You will have no problem at all and that's a beautiful drive.
Do watch your speed through Newhalem though (the area comes up suddenly). The popo have nothing else to do there and they are merciless about giving tickets.

Diablo down to Newhalem keep the speed down, it is crooked, and a good drop. but we do it every year. no problems.

WA SR 20 is a beautiful highway through the Cascades. Leaving out of Winthrop there is a couple Forest Service campgrounds. They maybe tight for your rig.
There is a very steep climb to the top of Wshington Pass. You should have no problem with you rig. Take your time and shift down. At the top is look out to Liberty mountain.
Next section to Rainy Pass is just a little down and then up again. No biggie!

From there west you will find a good but twisty in places two lane highway.Once reaching the Ross Lake Rc. Area there is a great campground at Thunder Arm.

The highway then drops down into Newhalem a Seattle City light town below the dam. West of town is another good campground.

Marblemount is the next town.
It has fuel.
You need to make sure you top off fuel in Winthrop as there is limited fuel along the route. Mazama might, but I do not remember any.

We will be coming from Jasper, Banff and Revelstoke, down the Okanagan Valley in B.C.

I've never been on Washington state hwy 20 before. That's why I'm asking.

I'm also thinking of finding a campground so we can explore the Northern Cascades for a few days. Not sure where to stay though. We'd have to find some place with hookups that can accommodate our 34 ft trailer.

Where have you been? It's hard to imagine getting to Osoyoos without having crossed similar mountain passes in Canada.