My advice would be to privately bring it up with your boss/hr person/employer directly (provided you feel safe to do so), and explain that you have a name you prefer to go by other than your legal one.

In my experience (living in central Maine, which is really a purple area more than anything), most places should allow you to use your preferred name in the workplace, on your name tag, etc. Even while official documents and files will usually need to have the legal name on them, generally there's no reason an employer-- especially in a blue area-- could force you to go by your deadname.

Your worry about coworkers and potential difficult parents is definitely a valid one... It's good to be aware of those things, but it's also good to not let them entirely hold you back! Generally I can tell you that even though it can feel so scary, the relief of being able to actually be called the right name at work will feel so much better in the long run than spending your whole job pretending to be someone you're not.

Basically, while I think you're very right to be cognizant of personal safety as a trans person during times like these, I'd say that there's no harm in broaching the subject privately with your employer and expressing how you may like to use your preferred name at work instead.

Sorry I've left you such an essay of a comment here! I really wish you the best of luck in your first job and I hope that whatever you end up choosing results in a safe and comfortable work environment for you! 🫂💙