I think I'd try Alaska also! Always wanted to see The Eagle Lady of Homer, Alaska and stay at Quiet Place Lodge We have never been to Alaska but it definitely looks intriguing.
Alaska would be great, but if you're locked into the lower 48, flying there with the free tickets isn't an option. You could fly to Seattle and take a roundtrip cruise from there.
Another option might be San Diego. My DH and I took a short (half-week) vacation there a couple years ago and really enjoyed it. Lots to see & do - the zoo is absolutely outstanding, Balboa Park in general is attractive and has other things (museums etc) to look at, the USS Midway (aircraft carrier) is permanently docked there & has tours, and of course there's the beach, Wild Animal Park (excellent), Sea World, and even taking a day to drive to Disneyland. Good weather pretty much all year.
If you're into history, try Boston (but DO NOT drive there!! Take public transportation; it's also a great walking city) or fly to Washington DC or Norfolk VA - Norfolk for Williamsburg/Jamestown/Yorktown, which can easily take a week to enjoy. Or Baltimore for the DC area, the eastern shore of Maryland (including Chincoteague/Assateague), Gettysburg & Valley Forge, PA Amish country, etc.
And even though it's too close to home for us, you might find Chicago interesting; tons of excellent museums (the Art Institute will astound you at what's in there!) Again, don't bother with a rental car - too much traffic & too expensive to park it; just use public transportation. During the summer, there's a water taxi that operates on the Chicago River between the Sears Tower area, Navy Pier and the museum complex; baseball games (choice of 2 major league teams!); shopping at Water Tower Place; the Navy Pier ferris wheel; dinner cruises on Lake Michigan, etc. Plenty to fill a week.
I like the south of the border option. Mainland Mexico has some beautiful places where you may not want to RV to - with flying the best option for a visit.
Couple years back, I was had a couple of United "sorry but this flight is overbooked" free tickets to use by the end of the year. I used them to get to and from San Diego, and booked a cruise to go along with a short San Diego vacation.
From San Diego you can cruise round trip to the Mexican Riviera in early fall (a week to 10 days), to Hawaii in winter (two to three weeks) or Hawaii and the South Seas (month and a half).
San Diego is good for a visit pretty much any time of the year.
Charlotte, North Carolina. Great region.
If you like NASCAR, visit the team shops nearby.
If not, take the easy two hour drive over to Asheville area to see attractions like the Biltmore Estate.
Oh, and congratulations on winning the airline tickets!
I would book a cruise and use the air fair to get to and from your port. I would fly a day earlier then your cruise leaves and stay over night in a hotel. You never know when you will be delayed and the ship waits for no one. I will say the Alaska cruise is great, we are going to do that again.
Los Angeles/San Diego/Orange County areas - we have enough stuff to keep you busy for weeks.
You can fly to Los Angeles and take a 4 day cruise (Mon-Fri. 4 day is cheaper than the 3 day which is Friday-Monday) that will take you to Catalina, San Diego and then Ensenada and back. The ship leaves out of the Port of Los Angeles (near Long Beach). They also do 7 day mexican riviera cruises out of the same port and we've done it twice - LOVE IT! That boat goes to Cabo San Lucas, Mazatlan and Puerto Vallarta. Royal Caribbean is the cruiseline i'd suggest over Carnival. The Carnival ones here leave little to be desired in food and service and typically are the "party" boats.
Once back in the Port of Los Angeles, you can drive to Orange County and do the Disneyland/Knott's thing.
Those right there are at least 12 days (depending on whether you go with the 4 day cruise or the 7 day cruise).
My other choice would be Cabo San Lucas for a week. LOVE it there.
If you want any info on the cruises or the L.A./Orange County area - feel free to PM me. We've done 6+ cruises out of the Port of Los Angeles in the last few years.
Central California/Napa area is one of the most beautiful areas of California - the Monterey Bay Aquarium, Hearst Castle, Wine tasting in Paso Robles (central calif.) or Napa (northern California).
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Exciting thought, isn't it? I missed the "already going to Washington DC" part too. 
