How to sync my iPhone with a new computer without deleting my phone's current contents?

I'm a graduating senior and just received a new macbook for college. With it, i made a new apple id so i wouldn't have to use the same one as my parents and siblings. However, i have an iPhone 5 that is synced with my old computer that i no longer have. I wanted to be able to use my iPhone with my new macbook, but it says that my phone can only be synced with one computer. It tells me that i can erase and sync my phone with my mac's iTunes library, but since it is a new computer, it doesn't have anything on it. Is there a way that i can manage my music and apps on my new computer without having to erase all of the data on my phone?

You can use some third party software like this Tunesgo, it can help you copy the photos from your iPhone to Mac for backup first, and then sync iPhone with your new account, re-sync the photos back to your iPhone. This method worked for me.