I used to worry about this, but don’t anymore. I simply carry a small backpack with ponchos for the family (less weight than umbrellas and not as bulky).
The rain can actually be a welcome relief if you go in the summer like we usually do. It cools things off and clears out the crowds. One year we went in August and it didn’t rain all week. It was hot, dry, and dusty. Then it rained two days in a row, and we were glad to have it. One thing that helps is that the rain is usually very intense (usually in mid-afternoon), but short. It usually lasts about 30 minutes, and rarely for more than an hour. Usually we can time it so that we’re in an inside attraction when it hits—we’ve only been stranded a few times where the rain was so intense that we had to wait it out (usually in Epcot for some reason).
P.S. The pools and water parks are usually a bad place to be in the rain, because Florida rain storms are usually accompanied by lightning, and if there is lightning within a certain distance they will shut down the pools and water attractions.