I'm not sure if you mean that you have a last name which is also seen a a first name (that's what I think you mean), or if you literally have two first names as you said (e.g. David Mitchell Smith).
Of course you should keep correctly them. Obviously, you could use your full name as a signature, but I wonder what would happen if you started to put a heading on your emails the way we'd do in business letters.
For instance, at the top of your work emails, you could put:
David Mitchell
Billing Department
Or, if you mean that you you have a last name that people think is a surname, do you have a middle name? If so, you could try David J. Mitchell.
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