English is not my first language, so I hope native speakers will correct me if I´ve misunderstood or overlooked the subtler differences between these words, but I´ll give it a go:
Awkward can be understood in a few different ways.
Referring to a situation/moment/happening it means "unpleasant, embarrassing" (It was awkward to see mom kiss dad)
Referring to a person it means "clumsy, unelegant"
These are the two most common uses.
To be embarrassed usually means to feel bad about your own appearance in some way, often because you have done something stupid and feel like a fool. A person can be embarrassed, while a situation is "embarrassing" (causing someone to be embarrassed).
Fx
I was embarrassed when I spilt my milk all over my teacher.
It was embarrassing to spill my milk all over my teacher.
As a verb: You embarrassed me when you pulled down my pants in front of the whole school.