To transfer files from the EVO to your computer and to transfer files from your computer to the One, you do NOT need HTC Sync Manager or any other software.
Just plug the EVO into your computer via the USB cable. The EVO will appear on your desktop as a USB drive. Open it, go to the folders where your files are on the EVO, copy them to the computer. Do the same with the One. You just plug it into the computer with the USB cable. The One will show up on your desktop. Open it, go to the folder where you want to put your music files. Copy them from the computer to the One. Done.
If your EVO or One are not showing up on your desktop, take a look at the phone's screen. There might be a message or notification asking you to confirm the connection via USB.
Regarding "strange music files" on the EVO that won't copy to your computer, you are more than likely not copying the real music file, but trying to copy some pointer file of some kind. Real music files will copy to your computer, no matter what format they are. (The only exception to this would be copy-protected files that were purchased and/or downloaded with Google Music or something like that, and that is clearly not the kind of files you have, and is outside the scope of this discussion.)