Can I add songs to iPhone via new computer while keeping the songs I already have on my phone?

Ok so my old computer just broke Rendering everything I have on iTunes useless, However I have all the songs I want stored on my phone anyway. Now can I insert my phone into a friends computer and somehow just add songs to my phone without losing the songs I already have stored on my phone, Also all the songs I have on my phone are NOT purchased from the iTunes store…

Hopefully there's a way so please guys let me know!

Try this:
Install iFunbox on your new PC.
Using iFunbox backup all your music files from Iphone
When the backup is complete you may now sync it back to itunes (itunes might/will erase your music library but at least now you have backup to sync it back)

Of course you can, from iTunes itself. This is not syncing. After you connect your iPhone to the computer, click on the device in iTunes, and on the top bar, there's music, movies, photos… And after all at the end there's a button "On this iPhone"
Click on that and on the left hand side you can see tabs Music, Movies, TV shows, and all. Click on the music button and drag and drop the music files you want. It is the way how I do. If On this iPhone button is not there, you have to use a third party app like iFunbox. But when using that, you have to make sure that the songs are in.aac format.
Hope this will help.