It is our tradition to go at this time every year (sometimes twice!). As the PP said, we are used to super hot weather, so low 60's is cold to me. In the AM and starting in the afternoon, we all start putting our jackets on. I always wear jeans with flip flops during the day, the have socks and Converse in the locker for night time. A few years I've even had a scarf while waiting for the night parade.
It has never rained when we have been. Always clear, blue skies, but that doesn't mean it won't be chilly. I have never brought a poncho to DL, and I never get that wet on Splash Mountain. I don't know about the wet ride in DCA, as we rarely go over there.
I must say, the weather we have had so far this year has not been typical. Way above average as far as heat goes all year long, and barely any rain. SO, you never really can say what will happen. But whatever does happen, have a great trip!