Leaving from Eugene, I would head to Portland and east on I-84 to I-80 and on east. The interstates are well maintained and All in the northwest feed the ports along the coast so truck traffic is heavy. The RV park at Boardman is open all year, it is a city park and reasonable. Further on there are parks that are generally open in Boise and Salt Lake. On through Wyoming and further east, I don't know about. I believe there is at least one near Cheyenne.
If you are headed further north, like MN etc., then at the Umatilla you can head north on 395 to Ritzville and then east on I-90. I would not go that way unless you have to as the further south you can stay, the better. If you do go east on I-90, let me know as we make that trip in the winter frequently and can give you a list of usually available parks.
One last comment, if you don't head east on either 90 or 80, go as far south as you can, i.e., I-40 or 10.