Sounds like the air/oil bath unit needs cleaning - if it's clogged (or overfilled) it will restrict air, causing excessive vacuum in the carb that will then cause the carb to draw in more fuel. Sounds like it may be introducing oil onto the air mixture as well - adding to the smoking effect.
You said you cleaned it. How well? Did you get the filter element out? Brake cleaner? Did you only fill the air filter bowl with oil to the line?
Other suspects - piston ring wear, allowing oil from the sump to be introduced into the burn. Have you compression tested the engine?
What did the old plugs look like? Can you post a picture of the business ends here?
1968 Series IIa
1997 Defender SW (Original Owner - Sold)