I think that the waterproof shoe covers are going to be very difficult to put on over your sneakers (my DH once ripped a pr. of galoshes trying to put them over sneakers), but if this is a real psychological issue for you, I guess you have to do what works for you. (How on earth do you shower if you hate to get your feet wet?) I recommend carrying a microfiber towel, spare slip-ons, and about 6 prs of socks in your parks bag. Maybe a roll of duct tape too, if you plan to ride Kali, because your feet will likely get submersed on that ride, and even with your galoahes, the water will get in from the top.
I hate wet socks, myself, but it's seldom an issue for me, because I also just hate socks, so unless it's really cold out, I never wear them. I live barefoot or in sandals from April to October.