Depending on whether you are one of those who like to meddle, you have nothing to lose.
If the printer doesn't discriminate between one and several sheets of paper, it is possible that the paper path and separator are merely dirty. Paper it seems, leaves dust which sooner or later will lubricate the pads and rollers which ought to grip it. That could well be a reason for the smudging too.
With nothing to lose, try careful dismantling until you can get at whatever is supposed to hold all the lower sheets and allow only the top sheet into the machine.
Then ideally some isopropyl alchohol on a tiny strip of cloth or cotton bud and "scrub away" at the rollers or pressure pads. If you havn't any "IPA" stay with water, a spot of washing up liquid, or even meths, but don't overdo it.
Now, if you have broken nothing, try putting it all back!
If it hasn't recovered you printer, you at least will have had an interesting time trying!
I have an Epson 800 which will go out with the bin next time! (It is squeaky clean, but I could do nothing about a dead segmant on the black print head)