Dave..... no, there really is no need to worry over copyright issues. Especially on original songs.
Most writers are not even close to the level where they need to be concerned over that....although we all think we are at the level where everyone is trying to steal our tunes..... nope it's not true.
As hard as it is to get someone to even listen to your music these days, having someone steal a song is so remote that it's not even close to being something to worry about.
I don't copyright my tunes. I use a 3rd party service called Songuard through MasterWriter2 software.
As far as a copyright goes, as soon as you record the song it's covered in a defacto copyright. that includes all the files and lyric docs associated.
But really, that is not something you need to worry about. If you have a few #1 radio chart hits and make nice royalty checks then yes, I would say that you are now in the class where that is closer to being a reality. But generally, those caliber writers are not hanging out here.
There are probably some programs available that will encrypt any file to keep it safe..... I'd test it on a scrap file first.