I have never used an ice auger before. But do know a few things about propane as a fuel. Propane is stored as a liquid in the tank, but pulled from a the tank as a vapor. That mean heat is needed for the propane to switch from a liquid to a gas. When it's really cold especially when dealing with a small tank. Sometimes it gets hard to get the change of state to occur. This is one reason houses use long tanks, just more surface area for the change of state to occur. In cars they run it through a preheater to help the change of state to occur. The tank is also pressurised which helps to keep the propane in its liquid state.

Now it gas it is of course stored as a liquid, pulled from the tank as a liquid, it too also needs heat to change to a vapor cause only vapor gas will burn. We have experienced a flooded motor before. But with gas they use the very low air pressure inside the venturi of the carb to quickly mix the the liquid gas with air to atomize the gas.

So in the cold even with a full tank, it could be possable to have a fuel issue. Now as I said my experience is with cars, they require alot more fuel to run than a small motor in an ice auger, so Maybe with the motor being that small, needing so little fuel to run, it might not ever run into that issue. But in cars, trucks, stationary engines, and even large btu heaters, it does become a concern.