My burner nozzle drips oil after it cycles off. We have replaced the filter and nozzle. We have also installed a new aquastat as the furnace was not restarting all the time.

You have a bad oil pump check valve. This valve is internal and not serviceable by the homeowner or HVAC service personel.

You can replace the pump with a new one

OR you can add a Honeywell or other manufactures oil line selonoid cut off. This stops the oil flow when the power to the burner motor stops thus stoping the drip.

One other possibility is a cracked nozzel holder. This can also cause oil to drip when turned off till the pressure bleeds out of the small oil feed line. (rare but it does happen)