The current date is already written in the text button. When you press it, a CupertinoDatePicker pops up, but I want to change the written date instantly when I change the date in CupertinoDatePicker. With normal date picker it is possible but after a few hours of work I couldn't figure out how to do that with CupertinoDatePicker.

In CupertinoDatePicker widget we have onDateTimeChanged method. you can update the string inside it. Here's an example