Are we talking blower operation or full heat operation?
Does the oil primary have a T and T for thermostat wires?
At your furnace where the reset button is.....this box has terminals where thermostat wires attach.....that is where (2) Two, T terminals will be....Some units have different format so I wanted to verify what yours had first.
That is correct. Now for why I wanted to know....
There are 2 reasons for it to keep running.
1) something is telling it to. In this case something would have to make the thermostat terminals connect. Things like thermost, wires touching etc.
2) something is stuck. The switch inside the primary control gets stuck after the thermostat turns off.
Given your symptoms I think relay stuck since you can tap it and stop.
When unit will not turn off...remove one thermostat wire from one of the terminals. (not the yellow ones that go inside)...if it keeps running it is a stcuk relay in primary control which means the reset box would need replaced.
2 questions, roughly how much is a new reset box and if this does not work but I accept your answer can I get additional advice later for this same fee
Depending on model $50 - $100.
And I never close my questions and always have follow up. Lets say you replace it a month from now....you can always come back and get follow up.
You will see from my feed back that I am not one to leave people hanging.
This may seem like simple banter, but I think every question through and provide the best answer I can....and when I am wrong, I keep moving forward.
If you get the model number off the primary I can be more specific on cost