Easy way to check the leak is first (brake fluid) and not axle grease, is to (if you are by yourself) dry the inside of the wheel hub and caliper cylinder along with the brake lines.
Get a new clean dry paper towel place them inside the wheel area under where the dampness was occurring most.
Fill the master cylinder and start the engine and pump the brakes a lot. A leak that bad will drip on the paper towel and you will know its fresh because its on the new paper towel. Then you have to figure out if its a hose or caliper seal.
Ya a shiny tire on the inside usually is a seal gone bad most times.