My 3800 started doing this after several years of use. Despite trying to talk with Epson about it via email for awhile, I was never able to get any better information than "take it to an Epson repair center".
Instead, I replaced it with a 3880, and it was doing the same thing from day one when I used the auto sheet feeder. I studied the problem, and noticed that the moving paper guide on the right hand side of the paper (it moves just before and after printing to help align the paper properly) was partially stuck, and not moving properly. (I still had the 3800 sitting around so I could compare what the guide should have been doing; it might not have been easy to see the problem otherwise.) Since it was under warranty, I called the Epson Pro-model-line technical assistance number (a brochure with it came with the printer), and they sent me a replacement printer in exchange.
It sounds like yours had some sudden mechanical problem along the same lines. I don't think there's any way to fix it yourself. (With my 3800, I asked repeatedly whether there was something in the printer I could adjust myself, and never got any direct answer on the subject.) You may need to call Epson or get it repaired. Hopefully it's still under warranty...
Lisa
