This is an awful lot of criss-crossing just for FP's. I'd go to DCA first to get the FP's, then head to DL, since you probably won't need to be back to RSR until later in the day.
Just for clarification, I think the Frozen Sing-Along show is connected to the rest of the FP system in DCA. At least there have been reports indicated that guests were not able to get that plus another FP right away. I know some folks think of this as an attraction and others think of it as a show.
As far as I know, as long as you've entered one park, you can pull FPs in either park. I know that in the past (I last tried this in November), my family would go into DL and then I'd collect the FPs and hop over to DCA to get RSR and/or WoC FPs and it wasn't a problem that their tickets hadn't been used to enter DCA that day.
I'm getting to old for this running! Good think my almost 14 year old knows his way around the park. What will we do when he grows up? We'll have to hire a runner to tour the parks with us! LOL!
