I did this once with my 330 formula. Exact same circumstances. Put it back in the water the following week and the AC worked just fine... Maybe I got lucky, it ran for a few days out of the water.

more then likely you have one of these pumps in the bilge:
LC-3CP-MD Magnetic Drive Pump | March Pump | March Manufacturing Inc.
if it was run dry long enough then you will need at a minimum just the impeller...but 99% of the time they need the rear housing as well. You can try and take the pump head apart and see if the impeller is just stuck on the shaft (i which case you can take a 17/64" drill bit and ream the impeller out and get going again) or if its fused to the rear housing. You can buy any individual component you want for that pump or what most people do is buy the wet-end-kit which is the complete new pump head with hardware and just swap them out. When you take that pump head off make sure its completely bone dry behind it where the drive magnet is mounted on the shaft of the motor...if you have liquid in there then you may have damaged the motor and in that case you would need a whole now pump assembly.
And if you haven't figured it out by now, i work for March pumps and get these kinds of calls all day long!

There's a chance the system generated a fault when there was no flow, and shut itself down before doing any damage. Usually it's a HPF (high pressure fault) due to lack of water flow across the heat exchanger. This same fault occurs when the strainer gets clogged.

When back in the water look for good flow coming out the through hull, if not then check the impeller as Wally said.

Both my boats have some type of sensor if there is no water and will shutdown.

There`s always a safety that will turn it off without water , the head pressure goes thru the roof.

You have a fitting "after" the pump for winterizing??? So you don't run the pump to winterize?? That may be the problem right there. That pump is liquid cooled...hence the "LC" in the name....if you don't run the pump you will not get all the raw water out of the cooling tubes...you may have antifreeze in the pump head but the cooling tubes may still freeze and rupture. I can tell when boating season starts in different parts of the country by the types of calls i get...most common one in the spring is "my pump is working but its peeing out these two holes on the side of the pump head where it looks like some plugs are missing" Yep we do have some small plugs on the side of the pump head that can act like freeze plugs but they are only there from the molding process and not to save the pump from a hard freeze. Best thing to do is let the pump push the antifreeze through the system to make sure you get all the water out of it at the same time.
Depending on how hard it is to get to the pump you can do one of two things to test it. First would be to leave it in the boat and get yourself like a 6-8' piece of 1/2" ID hose and disconnect the outlet of the pump (going tot he AC unit) and install your new hose right over the threaded 1/4" MPT fitting. Then turn the pump on and time how long it takes to fill a 5gal bucket...should take 1 minute. Or option #2 is to pull that pump out of the boat and dump it into a bucket of water. They are able to run submerged. Again, get a small section of 1/2" ID tubing and slip it over the outlet and time how long it takes to suck that bucket dry, it will suck your 5gal buck dry in 1 minute. And you should also be able to shoot a column of water 13' straight up in the air. If that all checks out then its something in the AC unit itself.
Another thing i have run into fielding these types of calls.....depending on how old the AC unit is, some had the high temp thermocouple/sensor mounted to one of the tubes on the AC unit with some sort of tar/goop type stuff and over time with the heat/cooling cycles it pulled back off the tube and is not making good contact anymore giving off false readings....may want to poke around and see if there's anything like that in the unit that may just need to be re-worked so it has a better contact again.