Welcome to the site !!
Some clarification is needed in a few areas .....
You mentioned it was dripping thru the "weep hole" , then mentioned it was "puking out the overflow" ..... So my question is .... Was it dripping out of the weep hole for the water pump , AND dripping out the overflow ??? , or was it one and not the other ??? , which would mean the "weep hole" you mentioned was the overflow hose ??
When you fired it up (I am assuming in reading what you said , that) , it was only stored for a few weeks , correct ???
Did you hear any popping out of the muffler when letting off the throttle or when revving it ??? (this would indicate a lean condition , which would cause overheating quicker than if it was not lean) , what type of gas was/is in it ???? , Non Ethanol Premium ?? , Ethanol pump gas ?? *still should be premium* , Race Gas ?? , Av Gas ?? ...... And how old/how long ago was the gas put in the tank ?? , Was the carb drained when stored ?? (the bike run till it ran out and fuel valve shut off , or the drain screw or plug in the bowl was loosened and all gas was removed *after the fuel valve was shut off*)
You mentioned the oil was "murkey" , like dirty oil that needs changed murkey ?? , or watery gray milky murkey , indicating a coolant or gasket issue ??
Was there any coolant on the ground when you moved the bike from the first location ?? , or after moving to the warmer location ?? ..... if so where was it located , near the water pump ?? , or overflow hose ??
Did you look over the Radiator Cap ??