Less than 3 months after my family and I occupied a rental house, we began to note mold growing on our personal belongings, in vents and on cabinet walls. A professional cleaning service indicated we "had a serious problem" which has already impacted our health. Our landlord paid to have a humidifier added to the house, but the problem grew worse. Now mold is growing more noticeably by the day. We have had sickness, had furniture and clothing damaged. We notified the landlord of our need to move out.

The landlord is resisting returning my security deposit, though no other damage has been identified in the short time we have resided there. What can we do to be sure our security deposit is refunded? Any state laws we can cite?

A humidifier? I hope a dehumidifier.

The landlord doesn't have to let you out of the lease and can sue you for any remaining months, unless the Health Dept shuts the place down.
You can sue the landlord for the health problems and damage to belongings, with receipts for medical care and pictures of mold damage (and hope you win).
A landlord cannot keep a security deposit for anything other than damage you caused to the apartment, with proof and receipts. This doesn't stop many of them from doing it anyway, and you have to sue in Small Claims.
These situations are often at a standoff, and you have to negotiate. Say you want the deposit back or you will sue for other damages as well as the deposit, and if he or she says he can sue for remaining months, you compromise.

Have you gotten the health dept involved? You should, because you will probably need their inspection to prove you need to break the lease.

Mold does not grow that rapidly, so the problem was probably pre-existing, but you need an expert to say that.