Does "I am getting cold as the night is drawing on" means
-"I am getting cold as the night is starting" or
-"I am getting cold as the night is finishing"?
Does "the night is drawing on" mean the night is coming to the end or the beginning?
It probably means that it will soon be night (understanding as to mean because). See this.
However, while looking for a definition as proof of my interpretation I also found this - definition 2. Oddly, Collins only gives the 'approach' meaning under American English, though I'm perfectly familiar with it as a British person. Indeed, it's the basis of a famous old joke.
"the night is drawing on" is a fixed phrase. It can mean "The night is getting near" or "it is getting dark" or it can mean "A significant part o the night has passed", probably with an implication that the night is seeming long to the speaker. Related to this second usage, it can also mean that so much of the night has passed, that the end of the night is approaching. This uses the phrasal verb "draw on", which has several senses, one of which is "approach".