From what it sounds like, you definitely need a new charging port. I would say it just moved if it was only the home button not working as the home button sits on top of the charging port and uses two pins there for its connection, but if it's not charging either then something else got ripped.
If you are planning to do the repair yourself, I would recommend buying the digitizer and the charging port. It's entirely possible to remove the screen without breaking it, but even the most experienced technicians accidently break them from time to time without meaning to.