The Breeze is not a two stroke, so oil would not be leaking out of the gas.
If it is smoking when it hits the hot engine, it is oil.
Have you taken the tank off to get a better view of the engine to see where oil is leaking from? Just a guess, but maybe the seal for the valve cover is leaking causing oil to spray out onto the underside of the tank, making it appear like oil is leaking from the gas tank.
Your idle is set too high, which may be the root of the reverse problem you are experiencing.
These are fairly easy fixes if you know how to find them....no offense meant, but you might be better off taking the machine to a professional mechanic