Yes, to answer your question, if you haven't paid all of your bills you can still switch your number. AT&T, Verizon and Sprint all require a working number as a part of their switch process to ensure you get your early termination fees paid for.
Even though you owe your old carrier money, you can still switch your number. They can't refuse to "let go" of the number.
However, it is probably a good idea to keep up with your cell phone bills or it can seriously hurt your credit. You should always pay off what you owe at the same time as porting out to another carrier.
If you are interested in switching, here's a great selection of carriers that will let you bring your own phone (BYO)