Best to let the floor dry completely to judge how dark they are. I would never soak the floor with any liquid (esp bleach), as you can cause damage to the wood fibers. If you use anything, try a pet stain remover that says it is okay to use on hardwoods.
When the stains have turned black, that indicates that the acid in the urine has actually damaged the wood. There isn’t much that can be done without replacing the boards.
However - sanding the floors will lighten the stains somewhat, and if you are lucky - the stains have not reached far underneath the varnish. I just had floors done with a few dark spots and they aren’t nearly as noticeable after the floor was sanded. Some of the spots are barely visible, and one is darker, but thank goodness it was a small spot.
You can also stain the floors after sanding, which will hide the areas even better.
Don’t ever paint or varnish a damp surface. Put heaters and fans on it if you want to dry it up. The exception is porous exterior stucco on a really hot day. Then you can spray it down with water, let any excess water evaporate or soak in (happens pretty fast on a hot day), and paint it with water based paint.