Paper loading issue, and cleaning cycle after every 2nd page.

I've inherited an HP Color LaserJet Pro M277dw with two issues.

First, if I put more than about 30-40 sheets of paper in the tray, the printer has trouble picking up a sheet. I'll get a paper jam message with the paper hanging down from the bottom of the printer (i.e., I remove the tray and there's a sheet of paper just hanging by its leading edge from the bottom of the printer). Keep the paper count in the tray low and everything's fine.

Second, if I print or copy a single page, everything's fine. If I print/copy two pages the printer ALWAYS goes into a cleaning cycle after printing the second page, and then prints the second page again. I therefore can't print/copy any document with multiple pages without getting duplicates of every page (except the first). If I try to print (for example) a five page document, I get page one, page two, cleaning cycle, page two, page three, cleaning cycle, page three, page four, cleaning cycle, page four, page five, cleaning cycle, page five.

Everything else about the printer works perfectly. I know there's no paper jam (or scrap of paper hidden somewhere) because I can print single pages all day long without any issues. I've already removed, cleaned, and reinstalled the rollers as per this guide:

HP Color LaserJet Pro MFP M277dw Rollers Replacement

Any ideas?