My experience of oily engines comes from a hobby of working on cars from the 1930's. Being about 80 years old at that time these engines were entitled to be oily, but mechanically they were surprisingly similar to the Jazz

Indeed one dodge was to replace the original 1930's pistons with Honda pistons ! (from a generator or something similar)
You are referring to spark plugs and pistons in the plural. Are you saying that ALL the spark plugs are white tipped and oily and all the pistons have oil on them as in the photo?
A rule of thumb before the days of ECU and emissions testing was that white plug tips indicates a lean petrol/air mix, black tips a rich mix or maybe burning oil , and milk chocolate coloured tip just right. So white tips with a petrol smell doesnt sound right. (although whiteish plugs are quite normal on modern engine which tend to run leaner and more fuel efficient than years ago. )
Also really oily plugs and cylinders will often be confined to just one cylinder due to maybe a broken piston ring, or damaged valve stem, as CRC says. With very high mileage all cylinders might get a bit more oily but usually one or two are worse than the others.
Its quite normal for the cylinder walls to be coated with a fine film of oil. Its what keeps the pistons lubricated, and why they have oil control rings. However if a fault is metering too much petrol into all the cylinders this can wash the oil off the cylinder walls, and form a thick petrol/oil mixture .It may be this coating the tops of the pistons and the spark plugs. It may look like oil but smell of petrol, or smell of burnt oil after combustion.
Unfortunately my 1930's based mechanical knowledge doesnt extend to knowing for sure this is the cause, or how to fix it. But I suspect if you can get the engine running properly and able to pass emissions tests it may as a bonus also solve the oil problem.