DAVdroid's purpose is to synchronize your DAVdroid account on the Android device with your CalDAV/CardDAV server. Your contacts in the "Local addressbook" and your events in "Local calendars" won't be synchronized by DAVdroid because they're not assigned to a DAVdroid account.
To move the contacts from your local addressbook to your DAVdroid account (and thus your CardDAV service), you may either:
- export your Android contacts from your local account (Contacts / Export to .vcf file) and import them into your DAVdroid account (Contacts / Import from .vcf file), or
- use software like Copy Contacts to copy/move contacts between accounts, or
export your contacts from your Android device and import them to your CardDAV service (not recommended).
Do not use the .vcf files exported by Android's Contacts app for any other purpose than importing again to the same Android Contacts app! Those .vcf files are using an ancient vCard format (vCard 2.1) and contain syntax errors. There are good chances that contacts will be corrupt when you try to import these .vcf files anywhere else.
To prevent using old contact information, it's recommended to delete the local contacts after you have transferred them to the CardDAV service.
To do so, you can – for instance – export your local contacts to .vcf as shown above, then delete all of them, and then import them only to the DAVdroid account.