When DS was in dipes I brought the cute diaper bag (we used cloth, so it was a big bag) and also a backpack for jackets and such.
At that time if we were on a big ride we used rider switch, so there was always someone to wait with the bags and boy. On other rides we just took them with us.
Last September we left some stuff in the bottom of the stroller, and DS's brand new pirate hat was stolen out from under a sweatshirt. So we leave nothing in the stroller anymore.
Recently I've had a shoulder bag for IDs, camera, and other essentials. Backpack is also carried for jackets, granola bars, and this trip maybe some real pb&j so we don't get tempted by the Uncrustables (aka high fructose corn syrup encased in bread) which are an impossibility for my family anyway.
I discovered in January that there's a nice little water refill area at Rancho del Zocalo (next to the salsas in the tables area) and also that you can just ask for ice water at a restaurant of any kind and they'll give it to ya. So we wont' carry bottles of water anymore, and we will definitely not buy any!
In Sept and Dec I had some sort of bag I don't recall, but now I have the "traditional" Disboards Baggallini messenger bag. Though mine is in the "on-clearance-at-REI" pink, rather than the traditional lime green. (just ordered a lime green cellphone to replace my "where the heck is it" phone, so at least I've got THAT bit of lime!)
Oh, for the days you will be at or hop to DCA, if your kiddoes are going to be allowed free access to the water play areas in bug's land, either bring a full change of clothes including underwear and shoes, or put them in swimsuits to play, then change back into proper clothes after. And bring a towel, too.