So, the mistakes were not getting the 3rd out by touching 3rd base before tagging the runner, and then leaving the field before appealing the runner leaving too early. Both are mistakes of the Nats not knowing the rule. You'd think that Adrianza could have focused on the important runner, the one who left 3rd early, rather than get into a race for trying to tag the runner from 2d before the runner trying to score crossed home. That's probably the simplest mistake.
The other thing that pops out of this is that the rule probably should be adjusted for when there's an appeal of the catch. It's normal for teams to leave the field if they think the appeal will fail. Unfortunately, they had to stay on the field through the without-merit appeal in order to then make the would-have-been-justified appeal of leaving the base early. It seems the 4th out rule should be modified for these double appeals.
