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So you really think the Kitty Hawk, with an FFG in train, can make better than 31 knots?

In reasonably low sea states (less than 4), and with no engineering casualties . . . absolutely.

Hull speed for an aircraft carrier is about 44 kts, and with their hull forms there's no reason they can't make hull speed.

I don't have the book in front of me so I can't remember the author, but I have "Introduction to Naval Architecture" (I know you can get it through Naval Institute Press.) which shows how to calculate top speed for simple displacement hulls. You can also see the calculation at:
http://www.sailingusa.info/cal__hull_speed.htm

The real question is how much past hull speed they can go. If you're not familiar with 'hull speed' you need to go look it up, but the short version is that it's the speed at which the ship's wave pattern approaches the ship's length, and in effect the ship is steaming 'up' its own bow wave. It takes power to do that (unless you transition to planing as a ski boat does). At speeds less than hull speed, the increase in power required for more speed is fairly linear and not particularly challenging.

Also through Naval Institute Press, you can get a series of books on the design history of US warships. The one on Destroyers says that a distinguishing characteristic of destroyers is that they can typically go faster than hull speed. Hence, a Charles F Adams class destroyer (with hull fineness ratio similar to the carrier - the Spruances and Burkes are relatively fat) gets about 37 kts on 70,000 horsepower (20 hp/ton) while a Nimitz class carrier gets about 45 kts (or so, it's classified and I don't know the real number) on 280,000 horsepower (3 hp/ton) because the destroyer has a hull speed of about 25 kts and uses brute force to get faster than that.

The Oliver Hazard Perry class FFGs are rated at 28 kts, but if you check the US Destroyers book you'll see that the US Navy rates ship speed with lots and lots of decrements. For example, it's at sea state 3, with hull fouling equivalent to 6 months at sea, with full fuel, weapons, crew and food, with engines nearing overhaul, etc., etc.

As a result, Perry's have been observed going 38 kts, which is way beyond hull speed.

If I were going into harm's way, in a transition-from-peacetime scenario where ships have not been in continuous combat for months, I'd make sure my escort ships could make a fleet speed of at least 30 kts. Fleet speed for TF 38/58 during combat in WWII was typically 25 kts, and there are multiple examples of carriers with half their engineering plant down due to damage yet still able to make fleet speed.

By the way, this is not all theory. While you're free to think I'm lying, I had a friend (since deceased) who was the engineering duty officer on a FRAM II destroyer sailing as escort on Enterprise when she was making her sea trials. At one point, Enterprise blinkered over to the destroyer that she was going to go make her speed run and she'd come back and get the destroyer later. The Captain called down to my friend and said, "I don't care if this ship blows up, we're not going to let Enterprise leave us behind."

They were up to 37 kts, with all the safeties bypassed . . . and the Enterprise sailed off over the horizon, then came back and rendezvoused with them later.

I don't know what the shaft pressure on a Nimitz at any speed, but I've been in the Engineering spaces of a couple of battleships, and the shafts are a bit bigger than those on your average pickup truck. I don't doubt each can carry its share of the 280,000 hp at which the ship is rated (see "Ships and Aircraft of the US Fleet").