Your very first assumption was correct.
The float valve in the carb has failed.
So you will need to fix this before you can do anything else.
When the carb is working properly then you need to drain all of the oil out of the engine.
Refill with cheap supermarket oil do one mow then change the oil.
If you bought a 2 qt oil bottle, change it after the second mow & the filter if fitted.
Fuel diluted oil will cause excessive wear in the engine very quickly.
Go to
http://outdoorpowerinfo.com/ , in the repair section you will find very detailed instructions covering carb cleaning .
FInd yours and follow the instructions to the letter.
Before you refit the carb, hook it up to a fuel tank and work the float with your finger to check it cuts off the fuel before the float runs out of movement.
Before you remove the carb, take photos of all the connections at both ends so you can hook it back up when you have finished.