Is my K-type thermocouple defective, or am I using it wrong?
The most likely cause of a relatively large error in that direction  (if the junction is actually at the correct temperature) is a double-reversal of extension wire. Since color codes vary by region, the most reliable way to identify the polarity (assuming you are sure the extension is type K) is to match magnetic to magnetic (positive) and non-magnetic to non-magnetic negative). 

Proper polarity extension or thermocouple wire must be run all the back to the measuring instrument. 

It is extremely unlikely that anything is wrong with a thermocouple- they are far more reliable and stable than IR sensors. If thermal contact is poor the junction may well be reading correctly, but be considerably cooler than what you are trying to measure- the leads draw heat out. 

Ideally, immersion length should be 10-15x the diameter of the probe. 

Edit: If you cannot achieve that try shielding the probe from air (make an insulated cup against the vessel) and running the leads against the vessel so they don't leach as much heat from the probe tip. You would probably do better with a bead thermocouple- at least you can visualize it more easily.