There doesn't really seem to be a German word with the exact same meanings and connotations as the English word "awkward". How would you translate it in a sentence like "He's an awkward person" or "That was an awkward situation."?
A situation can be unangenehm or prekär. Similarly a person can be unangenehm. Or schwierig, seltsam or unbeholfen.
I agree that there doesn't seem to be a word that captures the connotation of awkward 1:1. It depends on the situation.
That being said without knowing more about the context I'd translate the first sentence as
Er ist eine seltsame Person.
It sounds a bit weird though. Using schon makes the sentence a bit more elegant but being a native speaker unfortunately I can't give you a rationale for that.
Er ist schon eine seltsame Person/Persönlichkeit.
The second sentence could be
Das war eine unangenehme Situation.
Regarding a situation:
depending on the context. For general usage, I think heikel fits best here together with unangenehm as suggested by musiKk.
As for a person, "linkisch" seems to fit as well.
Somewhat of a false friend with English. It means 'funny' but in the sense that funny can mean weird.
"Schau mal diesen komischen Typ an!"