Wherever "let go of your hand" is used, can "let go your hand" be used in its place? Is there any difference at all?
There is nothing grammatically incorrect in your phrase.
Let your hand go.
It is the same, where "your hand" is the direct object of the verb "let". You've simply inverted part of the sentence, which has no bearing on the grammatical soundness thereof.
It does sound a bit odd, all the same, at least, to me (native US English speaker) it does.
In modern, informal (usually rapid) speech, a constituent can be placed at the end of a sentence for focus. The constituent can be one that's not actually licensed by the argument structure of the clause.
In the OP's example (archaic reading aside), your hand is not actually licensed. The phrasal verb let go takes a subject, and possibly a for- prepositional phrase as a complement. It does not take a direct object. But you add your hand for assertive focus. This construction is grammatical, but only in informal speech (including a written account of a conversation), not in formal writing.
Another example: Turn it down that noise. Here that noise is normally not licensed because it already fills the object position, but it gets tacked on to the end of the utterance for focus.