Unresponsive iPhone 5s touchscreen. Anyway to recover data?

I have an iphone 5s that was working fine until one day froze itself somehow. The screen is not damaged or broken but completly unresponsive. The phone though rings and receives texts and play music when I plug it in my car USB. So I would assume the logic board is not damaged as well. And since there is passcode on the phone I'm not able to connect the phone to itunes as it asks to unlock the phone with the passcode.

I thought it could be just a screen malfunction. So I took it to the Apple store for screen replacement but turned out that even with the new screen the phone is still unresponsive. They were not sure exactly about the problem or the phone functionality. So they offered me to replace the phone for a very reasonable price. But my concern is that some data has not been backed up for a month and would love to recover them.

I was thinking to disassemble the iphone myself and remove the logic board and put it in another working iphone 5s. I wonder if doing so would help me to access the data that was on my iphone??? Any advise?? And if not is there anyway to recover the data???