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>I'm looking for advice/tips/suggestions to get the most out of my time there with a limited budget.
Take 1 camera and 1 lens. I just saved you time and effort; no need to carry around a ton of crap and no second guessing which lens to use.
>How do I avoid ending up with nothing but tourist snapshots? Plan out shots before-hand I guess?
Don't be afraid to visit/shoot the tourist areas, but fill your time between visiting such places shooting photos as well. I shot lots between tourist destinations and typically put my camera away when I get to the really popular spots. (or shoot really loose to capture the touristicness get loads of context)
Every city has its spots geared towards tourists. This isn't the real city. Spend your time enjoying the areas that aren't designed to cater to you. Go where local people go and not just where Lonely Planet and Tripadvisor suggests.