Your location matters a ton here, but you're probably in a stronger position than you think. Most places recognize that a landlord has to keep a rental habitable - meaning safe and functional. Mold that's actively growing and spreading is a serious health hazard, and three months of written complaints with no action is going to look bad for your landlord if this ends up anywhere official. Document everything: take dated photos of the mold, keep copies of every message you've sent, and note when you complained verbally too.
Before you break the lease, try one more formal move that actually works. Send a certified letter (or email with read receipt, depending on your local rules) stating that you're giving the landlord a specific deadline - usually 14 to 30 days - to remediate the mold, and that if they don't, you'll consider the lease broken due to breach of habitability. Many landlords suddenly get moving when they see "certified mail" because they know you're serious. Some jurisdictions even let you withhold rent or use rent money to hire someone to fix it yourself, then deduct the cost - but check your local laws first, because doing this wrong can backfire.
The practical thing most people don't mention: contact your local health department or housing authority. File a complaint. They can inspect the apartment, document the problem officially, and sometimes that inspection alone lights a fire under a lazy landlord. Even if you don't break the lease, you'll have an official record showing the mold was there and the landlord was warned, which protects you if anything goes wrong later.