I'll just emphasize this: you do you on vacation. Want to carry a fanny pack? Do it. You know no one there. Go for it!
If you choose a fanny pack, I like North Face's small bags, which I've carried in the parks (as a cross body bag, and not as a fanny pack...even though it was a fanny pack.)
My go-to park bag (if going without DS3) is a small Coach cross body bag. Holds a small wallet, sunglasses and my phone.