There are several ways to give birds a drink.
if it is calm just dip the beak in water and let it drink naturally. Your method of dropping water of cotton wool is fine.
Dropper, syringe or drop-wise from a teaspoon is fine. Just a drop or two at a time. Don't force water into the bird, just allow a drop at a time to fall into the beak, let it swallow.
It probably only needs a few drops at a time, a pigeon will probably only need 50 to 80ml a day
If it is a racing pigeon it probably needs several days rest and some wholegrain / split maize.
Does it have a metal ring on it's leg with a code and/or number on it?