Pulling dbdata file with not rooted device
You don't mention which Android version is running on your device. With &lt; 4.0, chances are close to zero: without root access, you cannot access those places.

With 4.0 and up, you theoretically would have the possibility to use adb backup (see e.g. our backup tag-wiki). Thing is, you have to approve that from your device by tapping "OK" on the authentication popup – which is not possible with a broken screen.

TL;DR: It's not possible.