pat gave some sound advice.

I have stayed at both Horny Toad and Royal Palm. While I prefer Horny Toad, you are not really in a great place to not have a car. The beach is great as are the accommodations. Royal Palm would offer lots of places within walking distance but the beach is not that great. Staying at Royal Palm would allow you to walk Simpson Bay Beach (over the bridge) and get to a nice area.

Cars are not that expensive. Personally, I would rent a car and stay at Horny Toad. Good luck.