Most women do not experience mood changes severe enough to stop using their method of birth control, but that doesn't mean it can't happen.
It's rare to experience extreme mood changes on birth control, but if you're concerned, there are some single-hormone and hormone-free options to try. Progestin-only methods (containing only the hormone progestin) come in pills, a shot, an implant and even an IUD. There are also completely non-hormonal methods such as condoms and the ParaGard IUD, which is one of the most effective methods available. If you're concerned, though, check with your health care provider to find a solution good for you.