Hello!
We have a fairly old TV (brand name Classic?) that was bought overseas. We just recently moved and now we are hearing a fairly faint (but very annoying) static sound.
We hear the sound when using the television speakers, as well as when using external speakers. When we play a DVD using the external speakers, we have no trouble with the sound.
We do have a cable box.

Thanks for any help!

Sounds like line noise coming in from the cable. You can check out a local electronic store or Radio Shack for a line filter.