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.
What about merkwürdig in the sense of "somehow strange"?
Er ist eine merkwürdige Person.
seems to fit, as well as
Das war eine merkwürdige Situation.
And for me, merkwürdig has - phonetically and also regarding usage - the exact same connotation as awkward.
If you are in an awkward situation your feelings might be defined as Verlegenheit, e.g.:
Du hast mich in Verlegenheit gebracht.
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!"