continuity not working except for receiving calls, iMessages and AirDrop
This is so hard to troubleshoot - I wish I had some good tips. Calls and iMessage probably are relying on APNS push notifications and AirDrop happens on demand - so you might have some blockage or delay in the full continuity system.  


Are you able to take the devices to a location where WiFi and Bluetooth are not congested like an empty field or a house with one Access Point?
Are you confident Bluetooth works on the devices (and is ON)?
Are you confident WiFi works on the devices (and is ON)?


Sadly, the “sign out of iCloud and/or reset all settings / erase all content and settings” ladder of escalated troubleshooting on iOS is likely the way you’ll get this fixed. But, you might lose any apps that aren’t available for re-download or downloaded on iTunes (if you have older apps that aren’t for sale anymore). Also, that three step process takes a lot of time for devices with 60 GB of iCloud content or more.

Sorry for the “non-answer” answer but I wanted to explain there is no good consumer level debugging or logs that are publicly documented to get at the root of what’s happening with continuity.