Hello y Charles Catania,
It sounds like you're unable to drag a contact from All Contacts to a group within Contacts on your Mac. Also, you only started experiencing this after updating to macOS Catalina 10.15.3. I'm happy to help you get to the bottom of this.
Dragging contacts into groups in Contacts is still possible in Catalina. However, this won't work if it's a Smart Group you're trying to add to. Additionally, if you're using an Exchange account in Contacts those contacts can only belong to one folder at at time, and since they are automatically in the default Exchange Contacts folder, you may not be able to add them to another group. You can view more on all of this here: Add or remove contacts from a group in Contacts on Mac.
If you aren't trying to add contacts to a Smart Group and the information regarding Exchange accounts doesn't apply to your situation, restart your Mac and see if you experience the same issue: How to turn your Mac on or off. After that, create a brand-new user account on your Mac, log into that account, and test your contacts there: Set up users, guests, and groups on Mac. Finally, log back into your normal user account, create a new group in Contacts, and see if you're able to drag any contacts to that new group.
If you continue to experience the same issue, respond back with answers to the following questions:
- Are you using iCloud to sync your contacts?
- Were you able to drag contacts into groups when booted into the new user account?
- Are you able to drag any contacts to the new group you created?
Thanks for using Apple Support Communities.
Take care.