You don't say what month you're going to do this.
That matters, given the nature of your questions.
The lines, and crowds once you get inside, are probably going to be very long on the free day.
Reservations are highly recommended for both museums.
Reserve the Uffizi well in advance, several months ahead for a summer visit.
(I don't know if reservations are accepted on the free days; you probably should research that).
If I have to pick one, which one should I pick?
Many experts have said the Uffizi is the best collection of Italian art in the world.
If you're in a hurry, The Accademia could be viewed as a one art-piece museum: David. You can walk around the magnificent David several times times and really savor each view, in about 10 minutes, once you get into the museum and the area where David is.
However, even for someone in a hurry, I'd recommend about 30 minutes, there are at least another 3 Michelangelos there.
The Uffizi is comprehensive but so specialized that it's actually not enormous like the Louvre, The British Gallery, or NY's Metropolitan Museum of Art. So it can be seen in a couple of hours of quick walking. Rick has an mpg tour that might even get it down to less than 2 hours.
The problem for you is going to be getting inside, and then, on the free day, navigating through the crowds.