Eastbound, without the S is used like:
I'm Eastbound on Route 10. I guess it's used almost as a noun.
Eastwards is an adjective. As in "they are headed eastward".
Neither of these expressions are used very often. Much more common to say "I'm headed or travelling east"
That's not a great answer, sorry about that. We don't use it very often. Hope everything is well with you in Ukraine!