Broken iPhone 6 can't connect to iTunes
Sounds like the digitizer (the part that recognizes your touches) is also broken. My experience with Apple store is they will try to replace your screen first (at a cost). 

Many third party repair shops can help you get a working digitizer connected without erasing the device. Expect to pay them extra for the data transfer as part of the repair. You could even just pay them for the data transfer and then go back to Apple for a repair that has warranty from the manufacturer.