The iPhone can ONLY be synched to one iTunes library at a time.
If your new computer iTunes library is using the same AppleID as the library on the phone, do the following:
Before attaching your phone, make sure you've disabled automatic synching. Make sure the computer is authorized.
Attach your phone and transfer your purchased songs to your computer (In iTunes on your computer File>Transfer Purchases)
Then once that's done go ahead and synch. Yes, your phone will be erased, since technically it IS a new library on a new computer. However, once that's done, you just resynch with your new library which should contain all your purchased songs.
Note, however, that non-purchased music will NOT transfer from your iPhone to your new computer.
1. Disable automatic syncing on connection in iTunes Prefs > Devices > Disable...
2. Connect phone'
3. Right click device tab on left border of iTunes window and select "Transfer Purchases." This will transfer all iTunes Store purchased music, media and apps to the iTunes library on your new computer. it WON't transfer non-iTunes music or media
4. Right-click the device tab again and select "Backup."
5. It's now safe to sync with that computer.