Can't add music with new computer?

I can't figure out for the life of me how to add music with my new laptop. My old laptop broke and can't be accessed at all, so I have no way of accessing itunes on it or pulling off my old music. I have downloaded itunes onto my new laptop, deauthorized my old computer and authorized my new one, but I just can't figure out how to add music.

I added some new songs to my itunes library, can I not simply just add new music onto my iphone? It wants me to sync my iphone with my current itunes library which means it will erase everything on my iphone and make it match my itunes library, which only has like 5 new songs I just added.

I have an iphone 5s with iOS 8.2.

The only thing I can think to do is to download some 3rd party software to rip the songs off my iphone onto my computer. Then add them to my itunes library and resync this iphone with the itunes library. It just seems like such a backwards and long method though, am I able to just send new songs to my iphone?