Your right to repairs by your landlord does not authorize you to withhold rent. There are also several other rules governing your right to have repairs done on broken items in your apartment, as well as rules for what remedies you can seek if your landlord does not make necessary repairs. The Austin Tenant's Council lays out these guidelines.
Repairs are categorized as either ones that threaten your health or safety or ones that do not. If your dishwasher or garbage disposal doesn't work, then it is not a health/safety threatening repair. Texas law requires that your landlord: provides an apartment that is decent, safe and sanitary, repairs items threatening your health or safety, provides hot water heated to a minimum of 120 degrees Fahrenheit, provides working smoke detectors, and provides locking doors and windows, including a keyless bolting device.
In addition, Austin Housing Code requires heat that can maintain at least a room temperature of 70 degrees Fahrenheit, hot water supplied to plumbing fixtures at a temperature of at least 110 degrees Fahrenheit, and a kitchen sink, a shower or bathtub, a toilet and hot and cold water.
If any of these requirements are not met, you can request repairs, but you may not withhold rent. If you don't pay rent, you can be kicked out of your apartment and give up the right to have repairs made until you pay rent.
If your landlord doesn't get back to you when you make requests by phone or verbally, send a letter by certified or registered mail with the date, tenant's name and address, repair problem description, declaration that the broken item is a threat to health or safety, a request that repairs be made in a reasonable time period (a week for non-emergency repairs, 24 hours for emergencies), ask for a written answer if work can't be completed by the deadline, and a signature. If no repairs are made by the deadline, you can file suit in court against your landlord.
Hopefully, at this point your landlord will begin repairing your problem. But if they don't, The Austin Tenant's Council outlines further steps you can take, such as terminating your lease and moving out.