Shuttles serve hotels close to either airport but not both. As noted, the distances are great. Need to check the hotel Web sites. I don't think you can get a shuttle from DCA to even Rosslyn, but Metro works quite well. Cab from IAD to DCA will likely be expensive.

Since you'll presumably not get out of IAD until close to 11:30, after disembarking and getting your bags, you could stay near Dulles the first night (verify in advance that the hotel's shuttle runs that late; there is a Marriott almost across from the terminal), then take the Washington Flyer bus to the West Falls Church Metro rail station (uses the airport road, I believe, so traffic shouldn't be a big factor), and from there to DCA to get your car. Might as well wait until after rush hour, not just due to crowds on the Metro but to avoid traffic when you leave DCA (we don't know where you're going to). Besides, you'll probably get to bed late at night. If the cab fare of maybe $70 to a DCA area hotel doesn't phase you (maybe you can think of it as compared to the $400 you say you'll save by renting at DCA vs. IAD), staying near DCA would be the optimum, as you might not get to your hotel much later than if you stayed near IAD and you could take a leisurely trip to DCA the next day (by Metro if the hotel doesn't have a shuttle)..