It's grammatically acceptable but not not quite as natural to say "I began to be worried" as to say "I began to GET worried." You can also say "I began to feel worried."
"Be worried" expresses a static situation. The usual way to describe the transition into the state is with the verb "get."
It would be somewhat unnatural to say "I began to be worried/alarmed/scared/upset" etc. The usual expression is "I began to get worried/alarmed/scared/upset."
Another possibility is to use the verb "worry":
"” I began to worry
In this case the verb expresses the process of worrying.
You can use the verb "worry" in the past or present tense, as in
"” I worried/worry that you weren't/aren't getting enough sleep
...but that refers to a state that is not necessarily in effect at the present moment. For a state that is in effect at the present moment, you need to say "I'm worried."