Hi! I'm sorry this happened. It seems to have scared you a lot, which is totally understandable. As for advice, see below.
First off, please stop the bleeding. It's not a good sign for a cut to still be bleeding hours after making it. It massively increases the risk of infection and obviously blood loss in itself is not conductive to being physically healthy.
In order to know what care you need after you've stopped the bleeding, you need to determine how deep the cut is. Look up self-harm aftercare guides, they normally have drawings or pictures that you can compare your cuts to. From your description, it's likely to the dermis or hypodermis (fat) layer. It could also be deeper, but I can't tell because I can't see it (but you can). You generally can't (and shouldn't!) get stitches or other closures after 12 hours for cuts to fat or deeper so your best bet is to consult a couple of aftercare guides and clean and treat your cut according to the guide. For cuts to the dermis that gape open, they will scar slightly raised, and you generally don't need to use closures unless you want to. Keep an eye out for signs of infection and change your dressings at least once a day. If you plan on continuing to cut, consider keeping a first aid kit with you.
I won't tell you what to do regarding seeking medical attention because god knows I don't listen to that either, but it's reasonable to seek medical attention if your cuts won't stop bleeding, won't start healing, begin to look infected, or start to significantly impair your movement. If the cut is deeper than fat, or if you in the future cut nerves, tendons, ligaments, or vessels, try your best to seek medical attention.
With regards to being committed to a psych ward, learn what your rights are and try to find someone you trust who can help you with advocating for yourself. Seeking medical attention for any mental health issue can be very difficult, but having supportive people around you makes it substantially easier. Best of luck.