I can't sync new songs to my iPhone
From my understanding of the question, it seems to me that you are trying to sync your new iPhone with two different libraries.  This is impossible without a jailbreak (which by no means I am suggesting you do). 

Since you can't sync songs from your own iTunes library, it looks like you set up your iPhone on your mom's library. I would suggest that you select 'Erase and Sync' when you try to sync your music to your own library and allow your iTunes library to manage your new iPhone. Then, I would probably import your mom's music to your iTunes library.

You can do this using Home Sharing, as you said, but only if both of you use the same Apple ID.  Alternatively, you could follow this thread and simply share the organized files and put them in your ~/Music/iTunes/iTunes Media/Music/ if you're on a Mac. Hope this helps!