I know how stressful it can be around people - especially when we're not doing well ourselves. Please don't cut, you deserve better than that. It sounds like your boyfriend DOES care about you, but doesn't know how to help in the best way possible. Perhaps he should do some research into helping those who struggle with self-injury or cutting? I'm glad you've got a psychiatrist willing to help you out - but trying to find the right meds and the right dosages is only part of the battle.
Do you have a therapist/counsellor? If you don't - it's much easier to deal with self-injury (and related stuff, like depression) when we've got someone to talk to who is a mental health professional. Ask your psychiatrist for a referral if you don't have anyone right now.
Maybe you do need another stint in the hospital, but talk to your psychiatrist first. They'll be better to help you. If you can't see them SOON (like within a week or two) then yes, go to the hospital if it helps. It's not that you're not trying to get through this yourself, but sometimes we need extra support that only inpatient treatment can help.
You are not a burden. If you were a burden, your boyfriend would have left you. He's stuck by you, and that means he cares about you.
Get the help you need. You deserve it.