Hiya Jill!
For my great big, honkin' bag that I haul around Disney World I usually have everything organized into those clear zip top bags. Like all of the kids snacks in one bag, first aid items together in another bag, rain ponchos in another, etc.... That way, when security looks in they can quickly scan and not have to pick through bunches of little items and we are through the line even faster!
Another tip for getting you through security line faster is have your bags off the stroller and all of the compartments open and unzipped, or unsnapped, for security. It's one less thing they have to ask you to do then!
In late September you are still looking at evening temperatures in the low 70s and maybe dipping into high 60s. I was there last year during the first week of October and I used my cardigan during the evening and early mornings, but it was still really hot during the days! It's really hard to predict Florida weather, so I always recommend to bring layers for the entire family. Stuff some hoodies into the stroller! There have been times that we just knew it was going to be really hot and ended up purchasing beach towels to wrap around the kids in the stroller to keep them warm!
Have a wonderful trip!