I am unable to access my account settings from either the web interface nor the Android app. The "my account" dropdown does not do anything.

The My Account settings will not show in the Android app, but you should be able to access them via a web browser. When you log into your TurboTax account on turbotax.com, you should see, in the upper right hand corner, under My Account, Account Settings.  See attached screenshot.

The drop down does not work for me. Clicking on the down arrow does nothing.

Have you tried using different web browsers?  Which one are you using where the drop down isn't working?

Chrome and Firefox.