Most birth control methods that rely on the usage of hormones such as the birth control pill or implants can cause mood swings as they affect brain chemistry. The only birth control that absolutely will not cause mood swings are non-hormone type birth control such as condoms, diaphragms, or other barrier methods.
Different people act differently to different medications. There's no pill the magically will effect everyone's mood the same way. Be prepared to have to try different medications to find one that you feel comfortable with and works well for you. There's nothing wrong with having to try a couple of different ones to find one that's right for you.
A woman can only get pregnant for a maximum (to be overly safe) of 36 hours per monthly cycle. If you and your husband investigate natural family planning you will discover that it is as effective as chemicals and more effective than devices. You will be treating your body's ecology with respect and find that your marriage is strengthened.