Printer might have an encoder loss, stepper motor is dragging, or DOT Matrix print head guide rod is dirty ( gummed up )...
There is an encoder wheel or linear encoder that most of the newer DOT matriix printers use to position the stepper motor in the correct position when hammering the pins out into the ribbon which then transfer to the paper. If there is an encoder problem, generally the printers fail the initial homing and error out with red light and make loud clunk as the print head slams in one direction and the printer is alerted of a problem.
However if a minor encoder loss you might have where everything starts to get more and more offset as the printing continues as the intermittent encoder loss starts to build up and add to the margin of error.
Other times I have seen wavy lines but generally not a very large offset is when the stepper motor that drives the print head back and forth is worn out and it fails to keep a smooth linear sweep as it is hammering the pins out to apply ink.
Then the most common problem is the metal guide rod is scored with scratches or grimed up and causes the print head to not move freely, which causes wavy lines and larger offsets as long as the printer doesnt detect the offset error and error out. You can clean the guide rod with a soft cloth and rubbing alcohol. ( * Some printers require a mineral oil or Lithium Grease applied, which some run dry copper bearing on chromed steel for low friction. )
In addition Oki for example made some Dot Matrix printers that do not have a metal guide rod, but instead everything is plastic. Plastic Guides, plastic gears etc, and these generally experience problems like this when the moving plastic parts wear out and become sloppy. I think the Epson you have has the metal guide rod.
When cleamning the rod you will want to clean it to the point that no more grime is transferred to the dry or lightly alcoholed cloth, and be sure power is unplugged too when using alcohol which is flamable as always.
