Will jailbreaking my iPhone 4s allow me to copy songs manually in iTunes (or other software)?

I'm not an expert on jail breaking, but AFAIK, that limitation is baked into iTunes - it doesn't really care if your phone is jailbroken, at least for the purposes of your question - it just talks with the music app (or more specifically, the database behind it).

That said, there is software out there that can restore your iTunes library from an iPod or iPhone. iRip works on OS X, there's also DeTune. SharePod seems to be the goto option for Windows. You'll have to use one of those programs to get your library back on your new computer, then re-sync your iPhone from your new library.