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

Question

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.

Answer 1

I can see this is an old question but I had this problem this week and spoke with tech support for an hour + to get the answer.  The problem is visible within iTunes when you are connected to the mac and  try to sync music, but are unable because the screen says: iTunes Match is Turned On, even tho it is indeed turned off on your computer settings, and you have not renewed the subscription in the iTunes Store account settings on line.  The fix is on the iPhone or the iPad.  By the way, since this is an older post let me mention I am on Yosemite, and iTunes 12 and ios8 on the iPhone.
I tried several steps that didn't work, some that were listed on help sites and some the senior tech people had me try. 
The fix worked like this.  I plugged in the iPhone to the mac with the sync cable and with iTunes open and on the music setting it says, "iTunes Match is On" then I went to iPhone touched "settings" then "iTunes & App Store" then at the top of the next screen where it says Apple ID: in Blue  - touch your ID.  Then a menu appears that says, View Apple id, Sign Out, Forget and Cancel.  Touch the Sign Out Item.  Now you should be signed out of the iTunes store on your phone.
Next unplug the iPhone from the computer and plug it back in.  Now when you go to the section in iTunes to sync music it will say, "Sync Music Manually" and you can now sync as you want to do.  When you are done go back to the settings on the iPhone and sign back in.
I have an iPhone and 2 iPads, I had to do this procedure on each of my devices. 
The phone no longer has a toggle setting to turn off "iTunes Match" like the help pages say. I think the phone is supposed do identify that you are no longer subscribed all on its own, but if for some reason it doesn't, then this may help.