I highly suggest that if your kids are old enough to walk the distances involved, you let them walk, people don't pay any attention to strollers, wheel chairs or ECVs, I have seen people run into people in wheelchairs when the person in a wheelchair is off to the side and not even moving, then the person that ran into them starts cussing at the person in the wheelchair, the same thing happens with strollers, my DH was watching one of her nieces, who was sleeping in a stroller while the rest of her family went on space mountain, she was sitting at a bench with stroller turned so that it would take up the least amount of the walkway as possible, leaving enough room for at least 3 ECVs to maneuver by at the exact same time (to give you an idea of how wide the walkway that was still clear was). I would say at least 80 people bumped into the stroller, not one said sorry and I would say about 60% had a rude comment. Now mind you her niece was about 6 months old as I recall, so the stroller was a must. Now that being said, there are just as many rude people with Strollers, remember to be mindful of others, especially their ankles and especially when there are crowds, there are times that people just can't get out of the way. Ultimately you must do what is best for your child, just be aware of this and also remember that the child will be looking right at everyone's but all day, so if they are old enough to walk some of the time at least, you may want to encourage that just to give them a different view, I know I wouldn't want that view all day.
If you absolutely must use a stroller, remember there are a lot more hills than you think, but if you use them to your Advantage you can avoid a lot of the crowds and make pushing a lot easier.
For example at DCA, I recommend doing Soarin' First, then Grizzly River Run and the Trail, etc., then go across the Bridge to California Sreamin' and work your way down to Toy Story, etc., then go to Pacific Wharf and the Tortilla Factory and Bread Factory, then Bugs Land, then do the Hollywood Backlot, you will avoid about 80% of the really tough uphill climbs this way. (This is experience from pushing wheelchairs, I know strollers are lighter, but it will take it's toll on you by the end of the day).
You will not look like a dork, just be aware of all of the ramifications of having a stroller, a lot of them we never think about until we are in the situation.
Just my two cents worth on it all.