Domestic travel question

Hey, so I have a big question. I am a dual citizen of the US & South Africa. Im gonna be starting college soon in New Jersey, but I currently live in Chicago. My issue is that my US passport is expired, and I had applied for an expedited passport; however, I'm not confident it will arrive in time. I also cannot use a real ID to travel because my illinois ID is not the real ID version, and Im not sure I can apply for a real ID until I get my new passport because my birth certificate was given with my passport application and I cant get another birth certificate in time because I was born in New York so the process of geting a certified copy is at least a month. I do have my state ID, social security card, and my South African passport. Would I be able to travel domestically within the United States with my South African passport?