You can easily drive coast to coast without driving interstates (but not necessarily freeways) - if you are willing to use a mix of highway numbers, both federal and state (and even local). If sticking with one number, then your options are more limited. US50 is one, though it doesn't have kitsch and songs associated with US66 ('it runs from Chicago to LA ...', coast of Lake Michigan to the Pacific).
But one thing to be aware of is that most through highways have divided stretches between major towns, and many have limited access bypasses. Think for example of US31 and 131 in your part of the country.
The closest cross country route for you is US2. Across much of Minnesota and ND it is divided. In Montana is generally is not. In Washington there are a few divided limited access pieces.
Other choices for leaving the Washington coast include WA20 (joining with US2 in Idaho) and US12 (in Montana you many have to switch to MT200).
In Oregon, US20 is your best option for avoiding I84. And it will take you on to the city streets of Chicago. If you tried to follow US30 you'd have to drive I84 in a number of areas, but in Indiana and Ohio it is a mostly divided alternative to I80/90.
In the Saint Louis area US50 is cosigned with I44 and I64 and I255, though obviously you could find surface street alternatives. Kansas City(s) has the same issues. Across much of Utah US50 is cosigned with I70.