We are sorry to inform you that the adjective contrite means to feel regret, remorse, or even guilt.
Someone who feels remorse or guilt is contrite and in addition to feeling sorry, part of the definition includes wanting to atone for having done something wrong. The word comes from the Latin roots com- meaning "together" and terere which means "to rub." It's also related to the Latin word conterere and is defined as "to bruise." In the field of theology being contrite is "being remorseful for past sin and resolved to avoid future sin."