streaky printing?

I have an HP LaserJet IIISi. It's mostly Just Worked for me (though there was one mildly spectacular incident involving a failed electrolytic a while back). But recently (the last month or so), it's been exhibiting odd streakiness. Given the amount of collected wisdom here regarding such things, I thought it would be a reasonable place to ask to see if anyone can tip me off what's up with it and how I might be able to fix it. Specifically.... After leaving the printer off for a long period (a weekend, say), upon turning it on and printing, the first page is fine. The second page exhibits a few grey vertical streaks, worse towards the bottom of the page - not clean vertical lines; it looks as though toner is getting into the paper path where it shouldn't, and rubbing off on the page before it hits the fuser. Printing more pages works fine, but they get worse and worse. I've never had the streaks get so dark that I can't read printing for them, but the printing definitely becomes black-on-grey instead of black-on-white. Letting the printer sit will improve matters somewhat, but only somewhat; recently, I've even had the first page come out with some streakiness.