One of the things I have noticed with my kids is they HATE wet clothes. It they get a drop of water on them off it comes no matter what piece of clothing or where they are... with the exception of under garments.
There is not much you can do in these situations without a meltdown with our kids so I had to come up with a solution. The first thing is I have prepared our car and diaper bag to have an extra set of clothing for each child. I attempt to make them clothes that will quickly dry or feel that way. Sometimes getting something like a swim suit top or shorts will work in a pinch because they dry so quickly and you now can get them so they are a top and bottom that are more like shirts and shorts even for girls.
Okay so you don't have swimwear or extra clothes they can have in the car another way to quickly dry out clothes, if it's not too wet, is put it in the sun or if you live in a wet state like us you can place it by the heater with the wet spot closest to the heat. Even if it dries up slightly I have found my kids more able to accept the clothes back.
Worst case senario is because my kids are fairly young I have allowed them to go depending on the age and gender of the child and the place and event to go without that garment. If they're totally drenched and I have nothing then it's time to just go home.