I have read online that apple provides a $29 screen fix if the phone is under 30 day warranty, is this true?

The $29 price only applies if you purchased AppleCare+. See: iPhone Screen Repair & Replacement - Official Apple Support

You can purchase AppleCare+ up to 60 days after the purchase of the device but it would need to pass inspection. See the following from AppleCare+ - iPhone - Apple

Purchase AppleCare+ with your new iPhone. Or buy it within 60 days of your iPhone purchase:

• Online (requires you to verify your serial number and run a remote diagnostic)
• At an Apple Store (inspection of iPhone and proof of purchase are required)
• Or call (800) 275-2273 (requires you to run a remote diagnostic and provide proof of purchase)

It's free if it's a "defect" during the warranty. That would be more like a clean crack or a malfunctioning display. That should be 90 days for a replacement device or until the end of the replaced device's warranty - whichever is longer. If it shows signs of being dropped, then no.