"Abashed" as an adjective means embarrassed, ashamed, disconcerted. Unabashed, therefore, means not embarrassed or ashamed, and the adverb unabashedly means doing something despite it being potentially embarrassing or shameful. (It's also probably related to "bashful.")
So it's not quite an exact synonym for boldly, because you could do something boldly that wouldn't be embarrassing -- "boldly go where no man has gone before," for example. Yo wouldn't say the U.S.S. Enterprise was "not ashamed" to seek out new lives, new civilizations.
You might say, "He unabashedly requested cash contributions from everyone at the party." That could be a bold move as well, but "unabashedly" adds a bit of extra meaning, implying the speaker would find it embarrassing to impose on people that way.
And while we're on the subject, I hereby declare that the rule against splitting infinitives is silly, especially if you're splitting them with an adverb. "To boldly go" isn't the English-teacher-proper formulation. But "Boldly to go" sounds dumb and "to go boldly" is slightly awkward. In most cases, putting the adverb between the "to" and the verb sounds most natural.