Often common usage name changes are allowed in various states so there your free to use whatever name you chose. It's more complex if you need to fill out paper work for a security check or provide your social security number as that must match government records. As for job applications, hope that you will find a legal and preferred option on the form.
You might consider asking what the information is going to be used for as you will be able to discover if they are anti LGBT and if they are, you may not want to work there.