I like Millersylvania State park that is located just east of I 5 and a few miles south of Olympia. They have a lake, plenty of trees, and some sites with 30 amp electrical.
I would recommend Washington because of lack of humidity, and probably 20 - 30F cooler than Texas. If it is really warm near I-5, try moving inland to a higher elevation. Mt Rainier?
I took a trip through Washington near July 4th back in 98, and went to the Olympic National Park area, the only rain forest in the 48 states. It was very 'cool' in the summer. Lots of lakes and trees! I think that I moved around every few days, and one day I could have taken a ferry ride to Vancouver Canada, but skipped it because our dogs would not like waiting for our return a few hours later.
If you go north past Seattle, you do not need the Ferry ride, just drive in, enjoy and leave via I-5 or several other routes.
Going through Grants Pass (I lived there from 71 - 75) you might want to stop at Schroder County Park. It is south of town, to the west, and has plenty of trees, 30 amp electric, and grass to park on. IT is on the Rouge River, and you can see and hear the jet boats going by a couple of times a day.
The jet boats are a lot of fun, and go from town to Hellgate Canyon.
Hellgate.com They have dinner tours and daily runs in the morning and afternoon. In late August they might shut down because of low river water. They must have 6" draft for the larger boats, and a little less for the smallest ones. Get reservations if you want to go. It is a lot of fun!
There are several RV campgrounds along the Rouge River, between Medford and Grants Pass. It is about a 4 hour drive from there to the Washington State border. Traffic near Portland is hectic from 2 pm until about 7 pm, due to small freeways and hundreds of commuters. Avoid it on any weeknight or morning. Taking bypass 205 will speed things up and avoid the drawbridge on I5. Saturday and Sunday are still going to be heavy traffic days, but not quite as bad. I drive it daily, and it goes along OK.
Have fun on your trip!
Fred.