If you are using integrated security then you are logged in to SQL Server based on your Windows logon ID. That ID has to be defined to SQL Server as having access rights to the databases that you want to connect to.
Your first example uses Integrated Security on SQL Server. Your second example uses a non-Windows userid and password and NOT on SQL Server. So please decide what it is you want to connect to and how, and we'll try to help.
Let's replace the variables with tokens. For example, let's replace ServerAdressTextBox.Text with the string SERVER, etc, and see how your connection strings look at that point. Your three examples boil down to