yes, becoming adult is probably different from being independent. For some extent, I am non of those. I think being adult means being mature, which is something you can achieve quite soon or maybe a lot later. I consider myself quite mature, not too much but I feel I am not a teenager anymore lol. Truly becoming adult requires much time and I think it's truly achievable when you become fully independent.
So, what's being independent? From my perspective, is being able to do things on your own. I study in a different city but I don't have a job, so my parents are paying for it, which is kinda normal. However, I can do everything on my own: I can cook, clean, I know how to move in the city, which places are to be seen, I know how to organize my time and so on. Imo this is being independent: you can mostly do things on your own and if you can't, you are either able to find an alternative or you know how to seek for help and learn how to do things. I feel independent, but after all, if you don't have a job and pay things for yourself, you're not 100% independent yet.
But I understand how you may be worried and it surely isn't easy but in a way or another you'll find your way. May it be fast or slowly, you'll do it. So, my advice is just take your time and try to be open to new people and experiences