Hi Trimis, have you tried networking your old computer with your new computer?
For some help on how to do that take a look at this video,
https://www.youtube.com/watch?v=RYYe_Y-99fU
It's written for windows 10, but the principal is the same for whatever operating system you have, however some of the buttons you need to click are in different places, search for the location if you are not sure.
Once you have mapped the drive of one computer onto the other, you can just drag the files you want across.
Have fun.