The 2 "petcocks" aren't, they're valves, one hot one cold for low point drains. Close them, righty tighty, lefty loosey. Filling the fresh water tank is separate from connecting to the fresh water system. From your major leak description, you're probably hooking up fresh water supply hose to the fresh water system and it still has the low point drain's open.
From your description this is brand new to you, they could have been left open after winterizing, and haven't been closed.
Drain for the tank is located on the outside in the wheel wells, white plastic.