The way I read it is that you can still copy data with the USB cable, but not charge.

I do not know your exact situation so I cannot tell you exactly what to do.
With some assumptions based on your description, I would:

1. Charge the battery outside the phone. You can use a universal charger like this. (There are hundreds of similar or better ones, this was just the first hit I found.)
2. Get an SD card and copy your data onto it. Use the built-in File Manager. I had bad experience copying large amount of data with anything else - the watchdog timer can kick in and reboot the phone. File Manager avoids that problem. Then get the SD card out and copy your data from it using an SD card reader.
3. OR, if copying through USB works, use USB.
4. Whether you use an SD card or USB, I would watch the battery status and when it gets too low, stop the copying and go back to step 1.
5. Once all the data is safely copied, format the internal memory. A long press in File Manager offers the "Format" option.