My phone jack has a headache

Sounds like a wiring problem. You will need to figure out where the wiring runs and find any areas where wires are spliced together. There should be only one (usually), but you said this was an old house, so it is hard to tell how much it has been modified over the years. You may need to take the cover off of each phone jack and check for wires that may be loose or broken.
Ron-