How to add music to my iPhone with iTunes without erasing my old music?

So whenever I try syncing, I get this: "This iPhone is synced with another iTunes library. Do you want to erase this iPhone and sync with this iTunes library?"

And my only options are "Erase and Sync" and "Cancel".
I don't want to erase my old music so how can I add music without erasing my old ones that are already on my phone?

I'm sorry but the only thing you can do in that case is erase and sync

iTunes will not lot you sync automatically with the new library

So what you should do is

1. Do you still have the original mp3 files saved somewhere?

Then first add the new songs to your library

2. After connecting your device set it up as new device

Then go to device, click on music tab and check 'only sync selected media'

3. Press sync at the bottom

It's the only way to achieve what you want. Setting it up as a new device

Hope that helped

Actually before you set or put all your music on your iPhone you must Check the "Manually manage music and videos"
But since you don't, here are some steps:
1. At least have a backup of all your music on your iPhone on your PC
2. I recommend you use the DigiDNA Disk Aid an application for transferring files from iPhone/iPod to PC and its paid app (Google was always there for you to give you cracks and codes) http://www.digidna.net/diskaid
3. If you have DiskAid installed already and registered
4. Select all the music on you iPhone or Ctrl + A
5. Click Copy to Computer
6. Then a popup window will show up, it's all up to you
Note: Do the same thing for videos, it's a MUST because iTunes will erase the video too.
7. After the transferring finish open iTunes
8. Click on your device and find "Manually manage music and videos" and Check it
9. A Popup windows will show up. Click OK
10. Your Music and Videos will all be gone, Don't worry!
11. Find the folder of your music and drag it to your device on iTunes () Example pic of dragging to iTunes
12. Wait for it to finish syncing and DONE!
Note: Do the same thing for videos!

Please note that this only works with 1 computer so if you do it on the other computer you must do all the steps from step 1 again!