Pretty sure GM would be delighted to see water spots on saddles under dedicated riders working hard as opposed to somebody fretting about temporary, cosmetic discoloration due to being, gasp, used. Remember our traditional roots, Fox Hunters who routinely get wet and the days before there were indoor rings on every farm and we almost always showed outside.
Water does not damage saddles. It’s leather, a cow walked around in it for years. Improper drying-like leaving it the sun, a hot car or in front of a space heater will. Some tanning processes can result in permanent spotting but they are used in leather destined for garments, those processes are not used in saddles or any kind of strap goods except sometimes in really low end, cheap goods.
Just clean and condition as usual before riding, allow to dry out at room temp and condition after its dry…and it probably is going to still be damp the second day Any spotting will gradually fade and become part of that lovely, soft patina only lovingly used and cared for saddles show. Takes a few weeks or even months but it will blend in.
Resist the urge to over condition. You saturate that saddle with conditioner and it gets wet, it gets kind of gloppy and all over your breeches. Likewise trying to condition when it is wet just makes a mess- it won’t absorb very well. Never heard of the damp towel aiding drying unless it’s to slow it down if it’s hot and dry. Good air circulation around the saddle allows it to dry just fine. The leather may be a little stiff when it dries, just use light coats of conditioner and allow each to really dry before redoing it. Normal use also gets it back to suppleness with light conditioning after each use.
One thing to be aware of however is your leathers. If you are using the unlined all leather ones they will stretch if they get wet enough. The nylon/synthetic lined ones will not. So if you have the choice, the lined ones are worth the extra bulk.
Might want to look at your rein selection, some of the thinner, softer ones are hard to manage when soaked. And those black leather show gloves can bleed turning your hands black.