Well, since I'm not sure what US state you live in, I'm not sure of their procedures. Your best bet would be to go to your state's website and look up what's needed. If you live in one state, but were born in another, you'll need a birth certificate. Either way, you'll need a birth certificate, tbh.
Getting a passport really isn't as difficult as you'd think once you legally change your name. Go to the nearest Post office that does passports, and make sure you get TWO forms, in case you mess up on one, and just follow the instructions. I think the price is now $135, but I could be wrong. And yes, you'll have to send in your original papers, but don't worry, they mail them back.
It took my passport around 3-4 weeks to come back to me, and my papers a few days after that.
One thing that you and your Mom might consider is that yes, should you get stopped by the cops, until you have your new driver's license, you might want to carry one of the court papers showing your name change. I just folded one up and stuck it in my wallet, so when I was involved in a hit-and-run and needed to show it and my license to the cop, I simply explained that "I'm transgender and was in the process of changing my IDs over." He was very nice, but did tell me that I needed to get my new license ASAP, since its illegal in my current state to drive without a valid license.
Ryuichi