I can't sync music to my iDevice without it removing what is already on it

I have recently gotten a new computer, because my old one broke and we did not find the want to fix it just so it would break again (it was unreliable), and so now that I have new music on this computer that I would like to sync to my iPod and my iPhone it won't allow me to do that without iTunes first removing everything on my iPod/iPhone.

Considering that if I were to do this.. I'd forever lose all those files that are on my iPod/iPhone. I have been suggested to go back onto my old computer get all of the song files onto a storage device and transfer them to this computer, but the problem is.. my old computer is broken, and I'm fairly sure it's not even around my household anymore.
So, if there is any kind of way to get around this on iTunes or someway I can get my songs and such from my iDevices onto this computer, but not necessarily iTunes (first), it would be greatly appreciated.
Thanks in advance.