I have had this same issue with my iPhone 6. I tolerated it for months. I couldn't deal with it after the phone started selecting random destinations in google maps!
The phone would go unresponsive to touch, so I would lock it and start over until the problem happened again.
The problem seemed especially bad when composing a message. I had to act quick or my friends would get a message from me reading: b. B. bbb.b .
or other nonsense.
When I needed the phone, it could be unbearably frustrating.
Fixes I tried:
Updating iOS
resetting all settings (worked sometimes)
The problem would be really bad and then work tolerably... I put up with for too long and finally scheduled a genius bar service.
Then..... I looked again at my storage and asked myself the question, "Did this start when my storage became nearly full?..."
The answer is yes! I purged all of my photos, figured out how to get rid of the nearly 3 gigs stored in messages...
And voila! My phone works great again!
I could tell nearly instantly that the "feel" was back in my phone and it seemed more responsive than it had it months.
I'm tallying any goof-ups and so far, I've only had a brief freeze when deleting old voicemail messages, but that made sense and was fine again in a moment.
I looked everywhere for advice on how to fix this issue, but the obvious answer was right there.
The 16 gig iPhone 6 just has inadequate storage, especially if messages are stored "forever"
I've freed up 5 gigs on my phone and it seems to be working like new!