Seattle to Omak - North Cascade Highway
I used to make part of that drive about every weekend (Seattl eto Diablo) although I went 530 to 20, instead of up to 20 and then over. I personally think it's a more interesting drive. And there used to be an awesome shack place on 530 that served burgers, greasy fries and milkshakes to die for.
I don't know if you're interested in more than just food but Seattle CIty Light does tours of the Skagit Hydroelectric Project, which culminate in a chicken dinner. I worked for them years ago and when I got hired I thought "Who in god's name would find elecitricity production interesting?" The answer is, not surprisingly, retired people. But after taking the tour a few times, I can HIGHLY recommend it because the scenery is absolutely stunning. The chicken dinner is fine. Nothing to write home about but the whole package, I think, is really worth it. A little kitschy but overall, cool. Here's a link in case you want to check it out. (And a caveat that back in the day they only ran the full tour so I can't really say what the shorter tours are like.)
Unfortunately, the only town about halfway between Seattle and Omak is Marblemount, at about 90 of the 220 miles. There are a few restaurants there, though I've only had lunch at one (Buffalo-something or other). The next community, Newhalem, is a nice stop, but doesn't have much to eat except ice cream. Winthrop is much closer to Omak, but has a number of eating options. I was happy with a lunch at Grubstake&Co, eaten outside on their patio.
I make a point, when passing through area, of stopping at Hank's Grocery in Twisp. They sell their own smoked meats. In early June I was surprised to see they had expanded the store (now part of the 'Harvest' chain/group) to include modern features like a full service deli and on-site bakery. They even had a cheese selection that rivals some of the Seattle area upscale groceries (aimed at Seattle area vactioners?). Apparently the deli & bakery were making a big splash among the locals, giving the retirees, UPS drivers and sheriff deputies a new place to get lunch and donuts.