So, if it is only your charging port, the data is fine. How this will be fixed, I’m not sure. That might require the hard drive to be removed, but I’m not sure. If you have enough power in it, then you might be able to send them to the new phone (probably through airdrop, as it is the quickest) and that should be fine.
Otherwise, I would definitely suggest scheduling a call with Apple, and seeing what they can do. I actually did that yesterday as I had issues with my Mac, and I was talking to a real person in just under 10 minutes after I scheduled. I have never tried calling them myself though.
If worst comes to worst and it has to be removed physically, then I highly recommend going to Apple and getting it done. Sure, it will cost more than if you got the local phone place to do it, but photos and things of the like are priceless, so you would want the professionals to do it.
Best of luck!