If you want to go to the National Archives, just take the yellow line straight from DCA to the Archives-Navy Memorial stop. See the Metro map here. Then you can walk from there down the mall to check out the other museums and memorials. The Smithsonian stop is on the blue line, which also goes out to the airport, but takes a few more stops to get there (unless you get on it going east and change after one stop back to the yellow line).

This plan is quite workable. Given that you're arriving so early, you should be prepared for the fact that you'll be getting into DC before most of the museums are actually open (the Smithsonian system opens at 10 AM). So maybe you want to stroll the Mall in the early morning and look at the monuments. The Lincoln, Vietnam, Roosevelt, etc. are all out there for you to see. For this you should take Metro straight to Smithsonian from the airport and just walk west. See the monuments information here. For info on the Smithsonian museums, see here.

I'd plan to be on the Metro, at least on the platform waiting for your train, at 6:45 PM. This will give you plenty of time to get back for your flight without getting nervous. DCA is not large, and the security lines are surprisingly easy to get through.