Legal Permanent Residency: What to Do if You Lose Your Passport and Green Card

Hi I lost my South Korean Passport and also my green card. I have plans to travel within the next month. I have a deceased father who acquired US citizenship right around when I was 1 years old. I have his expired US passport. I recently finished my biometrics and am waiting on the physical card, but that will be several more months. According to some lawyers, I can apply immediately for the US passport? However, my mom doesn't remember whether my parents filed a marriage certificate. We used to live in Virginia and now I am here in Los Angeles California. In addition, I don't have a birth certificate. I tried to call the hospital that I was born in Korea, but they seem to have trouble locating my certificate or birth information. What is my best route of action here...?

I understand legally I am a US citizen but I have to prove it. But, seeing as I can't find my birth certificate or don't know where to find the marriage certificate for my parents I don't know how to proceed?

Legal Permanent Residency: What to Do if You Lose Your Passport and Green Card