Erase and sync on my iPhone?

Okay so i switched from a iPhone 4s to buying a iPhone 5s a few days ago. My iTunes library that had all my music on it was on my moms computer but I'm leaving for college in a few weeks so i need to transfer all my music to my mac. I backed up my phone on my mac and it showed all my music in my new library. I used to just use this free music download app and would never even use the computer to sync and download music and not have to worry about iTunes at all but the app i used completely changed and i can't seem to find a good app that lets me download songs that doesn't use sound cloud cause sound cloud sucks. So because of this i decided to try and do this all on the computer like i used to. I took a couple songs from youtube and transferred them to mp3 and put them into my new iTunes library on my computer along with all the songs i already had from my old library as well. However when i tried to sync the new songs, it said that my music was synced with another library and that it would erase and sync everything. I figured that i would be okay since i was able to back up my music from my phone onto my new library but i don't know if all my contacts and pictures are backed up and I'm still scared to do the "erase and sync" option in fear that it will complete wipe out my phone and i don't want that. So can someone tell me what to do? Or can someone tell me a good free music download app that has a browser that allows me to be able to go on mp3 skull and get music like that?

I don't know about the syncing/erasing situation but a good app that I use is musictube. I have tons of songs on that.