You may be eligible to file for adjustment of status without departing the US if you were admitted to the US in a "procedurally regular" fashion pursuant to case law. If you were not or cannot possible prove that you were, you would have to depart the US to apply for an immigrant visa based on your husband's petition. Note that you would face a 10-year bar due to your "unlawful presence" in the US, however, you may file fro a Provisional Waiver of that bar prior to departing the US as the "immediate relative" of a US Citizen. You should discuss this case with an attorney as soon as possible.
We need more information; did you have your own passport? Was any petition ever filed with your name on it? What country are you a citizen of and have you left the U.S. since you entered? You may qualify for Deferred Action for Childhood Arrival. You should definitely speak with an immigration attorney who can assess your case in greater detail and possibly find a path to lawful permanent residency. You may also want to wait for immigration reform to see if a path to citizenship opens up.
Please consult an immigration attorney. This does not constitute legal advice.
You definitely need to consult with an immigration attorney. If you came as a minor (especially if you came with your mother), you may have actually entered legally and that would make a huge difference in your case.
As for "chances," no good attorney will ever give you odds and/or make guarantees because your case depends on many factors, including the human factor (officers, sometimes judges), and that can never be accurately predicted. An attorney can tell you what options are available to you, the risks associated, and possibly give a recommendation, but the goal is to give you sufficient advice so you can make an informed decision and, should you choose to, take known risks as necessary.
The quick answer is that "yes" you can probably "fix" your status through your marriage, but there are many things (membership in groups, criminal history, etc.) that can make you ineligible for a green card/residency.
For these reasons, you should consult with an immigration attorney before you submit any applications with USCIS/Immigration. Consult with 2 or 3, similar advice would be solid legal advice, differences would be individual attorney strategy or incompetence. Select one you feel most comfortable with. If money is an issue, contact a local area legal aid or similar non-profit offering free/low fee immigration legal services. Absolute best of luck to you and your family.
It sounds like you will at a minimum qualify for DACA - depending on your age when you entered the USA - or a provisional waiver through your husband. You will need to consult with an experienced immigration attorney to determine your best procedure to follow. It may be that you can qualify for DACA so that you will receive employment authorization and be able to get a driver license. Once you get DACA you can pursue a provisional waiver or any other means of adjustment available to you.