I’ll try, but without “touching” it, it’s a pure guess work. I don’ have any idea about your furnace installation an location (up-flow, down-flow, horizontal, attic, basement, crawlspace), but there are a few things that might be causing that soot / dust.
If your new furnace has been installed in the same spot as the old one was, and the blower compartment is sitting directly on the floor, there might be a bottom closure plate missing (furnaces often come without it and installers forget to order and put one in – safety hazard). Without the plate, your furnace might be sucking everything from underneath and blowing it through the house.
Your old oil furnace could have a lower speed blower, or simple weaker one. After duct cleaning, there was still plenty of loose dust particles on the duct walls, and stronger blower might be picking it up. Also, air ducts might be too small for the new furnace.
One of your return air ducts might be open in some concealed space (attic, crawlspace)
I don’t think that this has anything to do with the chimney, unless its base (clean-out door) is open next to the open blower compartment… but make sure that the new liner is secured to the new furnace vent pipe, sealed at the top of the house brick chimney and you have a cap installed.
Let me know