Congratulations - you will have an amazing time!
Discovery Cove is an "all-inclusive" park so, no, you do not need to bring towels with you. You are also provided with sunscreen, shampoo, conditioner and body wash.
The lazy river is amazing but you do go under two "waterfalls" on the way around. I don't think there is any way to avoid getting your head wet, unfortunately. But if you can stand it at all, it is well worth it. As a glasses-wearer, I am not fussy about it either but I consider the lazy river a "must-do"!
Whether or not to wear water shoes is definitely a personal preference. I always wear water shoes but my daughter can't stand them. There are definitely rough spots in the lazy river where you can scrape your toes. Having the water shoes also allows you to get out and visit the aviary mid-river. I don't know about you but I am not a fan of walking through bird poop in my bare feet!
Any other questions, feel free to ask. I wrote an Aquatica "sticky" post a while back and am finally getting around to writing a Discovery Cove one now!