can I either copyright or trademark a professional modeling name

Yes, it's possible to trademark (actually, service mark) your name for your modeling and endorsement services, as many celebrities do. See an IP lawyer for help to make sure you cover all the services and goods you're providing and expect to provide.

Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.

Personal names (including pseudonyms) function as trademarks ONLY if they identify and distinguish the services the individual provides and not merely to identify the individual.

Golfer Lee Trevino was not permitted, for example, to register LEE TREVINO as a trademark because he was using his name merely to identify himself rather than as a mark to identify and distinguish any services rendered by him. In short, personal names do not serve as trademarks to "brand" people -- they're used only to brand certain identified goods or services provided by people.

So ... far more important than properly filling in a trademark application form you need very specific guidance from an intellectual property attorney as to how you must use your "professional modeling name" in order to develop the trademark rights that you may then -- and only then -- seek to register.

I agree with my colleagues and urge you to speak with an intellectual property rights attorney before proceeding.