When I've done it (and other hikes like it), I've just worn running shoes with hiking socks. The running shoes have lots of mesh and let the water drain out easily, and generally have pretty good support and traction. The hiking socks (wool or wool blend) don't mess your feet up as badly as cotton when they're wet and dry out faster. I've never had waterproof socks, so I can't speak to those - but in June, you're gonna be sweating a lot (it's hot - and my feet sweat a good bit), and you may have as much water inside the waterproof socks as outside. Sandals provide excellend drainage, but usually just prove to be a pain in the foot, as small rocks, sand and plant bits get in them too easily and irritate your foot, even with socks on.
I take light weight sandals along to wear in camp at the end of the day - that way my feet can dry out well at least then...
It troubles me that these days no matter how cynical you become, it's never enough to keep up
- Lilly Tomlin