It comes down to context, yes, but it's usually quite obvious.
When describing things in a literal sense, it will always mean 'full'.
It means 'drunk' when relating to people, and other beings than can become intoxicated, when used on its own.
If you see it in expressions like "full av X", it translates to "full of" or "filled with" even when describing living beings:
"Hun var full av liv."
"She was full of life."
Kinda off topic, but what are the etymological routes of "full"? in the context of being drunk. I kinda assume it is almost like a shortened version of saying "Full of drink/alcohol" that became commonly used. Like in English, that has dozens of words to mean drunk that generally have other meanings (smashed, wasted, pissed, etc.)
...Although having seen Norwegians drink, I could believe they don't feel "full"/"complete" until absolutely drunk.