Is it okay to start a new relationship before the old one has ended?
Why not end this relationship first, then?
I can't wait any longer, I want something that makes me happier immediately.
Then I think what you need to do is tell your current partner that you're no longer happy and to move on.  If you're still in a relationship with them, you shouldn't be trying to get with someone else.  That's cheating, which is wrong.
But I can't live without sex.
If you don't want to hurt anyone, then you should consider breaking it off. You seem like you have a sex addiction as well. You should seek help to get that under control before it becomes overwhelming.