Those previous answers are on the right track but miss the timing issue - three weeks is already well past the point where you should've escalated this. Texas Property Code §92.008 requires landlords to make repairs in a "reasonable time," and in summer heat like you're dealing with, courts generally interpret that as days, not weeks. The certified mail route is solid, but you don't have to wait around hoping it works.
Your best move right now is to call your local housing authority or tenant rights organization (most cities have one) and ask about your specific options. Some jurisdictions let you "repair and deduct" from rent if the landlord ignores a formal notice, though you'll want to verify that applies in your area before going that route - it can get messy legally if done wrong. You could also file a complaint with your city's housing department, which sometimes lights a fire under landlords faster than anything else. Meanwhile, keep every communication in writing going forward: no more phone calls where there's no record. Send one more email today laying out that the AC has been down for three weeks, you need it fixed by [specific date one week out], and if it isn't you'll be pursuing other remedies. Keep it businesslike, not angry.
One thing folks don't always realize: if your landlord is being this unresponsive, there's probably a reason - maybe they're in financial trouble or just don't care. Either way, you might need to decide whether pursuing legal remedies is worth it versus breaking the lease and moving. I've seen tenants win small claims cases for rent abatement during uninhabitable periods, but it takes time and effort. That said, don't let them push you around. This is literally a livability issue in july 2026 heat.