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Seven days is a fair amount of time for the trip you're contemplating, so your plan to go be a more northerly route is certainly feasible. So, make it fun and scenic from the start, and rather than heading south around Lake Michigan and slogging through the traffic nightmare that is Chicago, make your initial direction north, up US-23/I-75, Across the Mackinac Bridge to the UP. Head west on US-2 past Apostle Islands National Lakeshore. At Duluth, use I-35 and MN-23 to cut southwest across Minnesota to Sioux City. A quick visit to the Corn Palace in Mitchell, SD; a free glass of water at Wall Drug; through Badlands National Park; another quick stop at Mount Rushmore; a short side trip to Devils Tower off US-14; and then at Buffalo, WY stair step southwest on I-25 and I-80 to Salt Lake City. From there I-15 takes you by both Bryce Canyon and Zion National Parks as well as through Las Vegas on your way to Los Angeles.
Now that is a lot of driving, about 6 days worth, but it will take you through some very scenic country and will let you spend a few hours anyway at some of the best sites along the way. Trying to stay north until you hit the coast will chew up your seventh day with driving as well, even if you take the 'short' route through Chicago, and you still wouldn't have time to take the PCH which is a relatively slow 2 lane highway.
AZBuck