How do I unsync from an old computer and sync to a new one without losing data or actually syncing.

I have an iPhone 4S. I have always had it "synced" to my family Mac on my own user account, but I kept the "manage music manually" box checked, so it was never actually synced. I recently got my father's old macbook pro. I did not wipe it, but changed all the account information to my own. I plugged my phone into the computer and it would not let me click and drag songs into it. After a little bit of digging I found some settings that would allow me to fix this. However, when I tried applying these a message that said something like this appeared, " Do you want to sync with this library? Your device is previously synced to another library. Syncing with this library will delete all your data. Do you want to continue?" This isn't the exact wording but pretty close. So I tried troubleshooting that for about 3 hours. Eventually, I tried deleting my user account from my family mac entirely through the Admin account. I then tried plugging my phone in to my macbook, and the same thing happened. I have tried everything I have come across except actually backing up wiping and restoring my iPhone. I would prefer not to do this unless there is some way to get every piece of data on my macbook from my phone, as it has music from various libraries and I am not positive it will load everything. Is there any way to fix this?