Hmmm... I'm not familiar with that particular scanner/printer, but as someone who used to work on copiers, printers and faxes for a living, I have a few ideas.
"Vertical" lines are good. I assume you are defining "vertical" as "parallel to the direction of scan," that is, lead edge to trail edge. Good. That tells me that the optics and the frame that holds them aren't hopelessly misaligned. If, for example, the trail edge of the printed document was narrower or wider than the lead edge, there would probably be nothing you could do except throw it away and get a new one.
"Horizontal" lines, those perpendicular to the direction of scan, are displaced by several pixels on one side relative to the other. That suggests to me that one side of the scan carriage is dragging. If the scan carriage has drive cables only on one side instead of both (most are only on one side), I'd expect the problem to be on the side without the cables. Depending on how it is laid out, it could be something very simple, for example idler wheels on the non-cable side so it will roll freely to keep up with the driven side. If dirt gets between the wheels and their axles, they drag and that side of the image "lags".
Suggestions:
1. If your printer has a document feeder, use it. Often the scan carriage will stay stationary and just scan the document as it goes by, if there is a document feeder in use. This doesn't actually solve the problem, but it does bypass it. If you use the document feeder, clean the platen glass with furniture polish, not with glass cleaner. Glass cleaner strips the surface and makes it try to grab the document. You want the surface of that glass to be slick, so much so that if you toss a sheet of paper on it, it skates right off the other side. Rain-X windshield treatment will do the same thing.
2. If that's not an option, remove the platen glass and clean all accessible wheels and rails in the scan carriage with a mild soap and water solution and dry them thoroughly. Don't use alcohol as it makes synthetic rubbers brittle over time. In particular, check idler wheels to make sure they spin freely. If they don't, a drop of a light oil at the interface of wheel and axle may clear the grit. Sewing machine oil is a better choice than 3-in-1 oil, as 3-in-1 has a dissolved plastic of some sort in it that creates a coating when it dries. CRASH PRIORITY IMPORTANCE: Do not loosen any screws with a paint spot on the screw and the area around it. These are screws pre-set at the factory to keep the image square between the original, possibly several mirrors, and the part of the machine that actually registers the image (usually CCDs). If you loosen them, you'll never get the image square again.
Hope this helps. I can't tell you much more than that without knowing any more about the particular machine and/or seeing the symptoms myself.