If you want to treat your fish with the most care remove them from the dirty aquarium before you disturb the gravel. There is all kinds of stuff in the gravel that could possibly harm the fish when it is suddenly mixed into the water. (bacteria, toxic gases).
In reality you could just put the fish into a bucket with an inch or two of your aquarium water and they would live just fine as long as they did not get cold. But that would not be very respectful now would it?
You only have a few small fish so it won't be hard to put them into a small container for a while. Here is a basic process we use to acclimate wild caught fish for aquarium life.
For you few fish....
Prepare one quart of room temperature water that is chlorine free and put it into a non clear plastic container so that it is about 1/2 inch deep. Add one quart of water from the aquarium where the fish are now to make the water level about 1 inch deep. Make sure the aquarium water you add has no food particles or debris in it. This should bring the temperature and pH fairly close to what the fish are in now. Add a big pinch of salt. Put the fish from the aquarium into the container and cover it so the fish are in the dark. Set the container aside so the fish can rest undisturbed. The next day change the water the fish are in but this time don't mix it with aquarium water but still add a pinch of salt. Let them set another day in quiet. DO NOT feed the fish during this process. The next day change it again and don't add salt. They are now ready for their new tank and they should be snail free. The fish could live in this quarantine for well over a month as long as it is quiet, dark and you do not feed them.
Once you have done the above process the fish will have acclimated to new water and completely purged their systems of all food and snail eggs they may have eaten. You could now put them into a quart sized plastic bag with 1/3 water and 2/3 oxygen and they will live for at least 72 hours. If you put each fish in a small bag by themselves they would probably live a few weeks.
To get rid of the snails replace your gravel with new snail free material. Sterilize the tank and all objects you will reuse. Bleach works well for this.
To eradicate the snails from your plants is more difficult. While you have them out of the aquarium is a good time to treat the plants with a chemical to kill the snails. It takes some effort and time but it should be worth it. Put the plants in a container (not the aquarium). Check to see what nasty chemical the local fish store sells to kill snails but won't kill plants. Treat the plants in a container with a strong dose, no worry about the fish because they are not with the plants. Copper compounds like Malachite green should kill the snails, but make sure your plant species aren't sensitive to copper.
Potassium permanganate or formalin might work to kill the snail eggs.
It would be a good idea to keep the container of plants under your lights.
Once you are in your new house just put the aquarium back together and hope you killed all the snails.