Simple distillation & azeotropes
By definition an azetrope is a distillate that boils at a constant temperature, but is a mixture of two or more compounds. So you can't purify an azetrope by using more distillations.

Pure ethanol has a bp of 78.37 °C and pure ethylene glycol has a bp of 197.3 °C. The bp of the azetrope could be (1) below pp of ethanol, or (2) above bp of ethylene glycol. In theory the azetrope cannot be between or equal to either of the boiling points of the two pure chemicals. (The problem with theory is experimental error....)

I can find no evidence of an azetrope between ethanol and ethylene glycol using Google. So I very confident that no such azetrope exists. 

You noted that the distillate temperature was about 67 °C as the pot started to boil. You also noted that you read the thermometer when it began to produce the drops of distillate. 

Look at the drawing below. 



The glassware setup has two thermometers. 


Thermometer-1 is in the pot.
Thermometer-2 is in the pot.


Ideally the pot temperature and the distillate temperature would be the same. However when the solution starts to boil the distillate is just starting to contact the thermometer. The thermometers have some thermal mass, so it takes time to get them up to temperature. 

The distillate thermometer is relatively slow to equilibrate. So long as it is dripping distillate solution then it is colder than the distillate and will not give an accurate temperature reading. 

So the only explanation for this anomalous result is that you made a small mistake taking the reading.