I have the same problem and am using a brand new Apple headset. Researched Apple forums for hints and do see other threads about this. I've tried just about all the suggestions but the only one that seems to work for a short period of time is the cold reset with pwr+home. Problem returns after a short period of time. Not sure if it's software/apps incompatibility or hardware, or a combination of both...
So I chose this forum thread to post and opened a case with Apple Support: Case#43320343
Apple tech support is requesting that I do another restore but this time "restore as another user" meaning that I won't be able to use iCloud restore. Make sense so I will be backing this up locally on my iMac at home later this weekend and will update.
Thank in advance for any constructive input.