HP1022N laserjet printer there are black specs along the printed page. looks like toner smudges.

Problem is either the cartridge or the fuser.
Put paper in the manual feed tray(tray 1)
Print something. When the paper gets 1/2 way into the printer, open the
top cover. Pull out the toner, look down at the paper.
If it has the smudge replace the toner. If no smudge, replace the fuser.

If this just started after you replaced the toner cartridge, one would have to say that it is a bad toner cartridge.
Print something that smudges. Pull down the front cover and look at the cartridges. Pay attention to the top one(black).
Look at the drum on that cartridge. You should see the toner sitting there. Compare it to the other cartridges which should be clean.

If you have BLACK Marks on the paper the YELLOW roller does not need to be cleaned usually.

To clean the yellow toner off the roller use a DRY lint free cloth. For this issue my suggestion for a resolution to the Yellow toner issue is replace the toner. They are not supposed to leak in the first place.

As for the BLACK tone on the left of the page you most likely have a Fuser issue causing that.

remanufactured / compatible toner? sounds like defects to me. try OEM, or another supplier

The fact that it's printing anything suggests that it is not a Toner problem. Remove the toner, and have a look at the run inside the printer, checking for any excess toner which is in there. Also, check the Drum and Imaging Unit for damage, wear and tear etc. There will also be a static strip near the imager, with a little tab which you can sweep across the unit to clean it.

try replacing the toner cartridge if that does not help it may need a fuser replaced

More than likely you have a bad toner cartridge.

Inkworks
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replace the toner cartridge, you wont have anything "dripping" out of it as the toner inside is a powder. if it still appears replace the fuser.

Its probably a defective toner. If the film on the toner drum is scratched it can cause toner to stick to the drum everytime it spins causing the constant marks.

When the paper is fresh out of the printer, does the ink that looks okay smudge if you rub it with your finger? If so, it means the fuser is going bad. Otherwise, it's probably a bad toner cartridge that's leaking toner onto the paper.