Some Thoughts
I am not an "Carburetor " man and don't know if the bike has been parked. for longer time
So if no vacuum line the bike has been running in an open position.
And on my 1980 there is such a position pri
others are on and res
So the petcock has been open before but not in the res position.
The float valves stop the flow so it does not need to be a big problem
I once modified a stock petcock .. And removed the handle with front
and put an aluminum plug with some
hard loctite .. Cant exactly remember how I did but had i so for several years
At least five.
What has happened I believe is that dirt ( rust ? ) at the lower level from the tank .
Has come down and the float valve does not close.
That can happen fex at startup in the spring. but stops after a short time.
Have also happened out on the road one time . ( The girl passenger looked Nervous )
It drips right under the carburetor that have a hanged float
I tap gently on the outside of the float bowls. with a screwdriver handle och wooden hammer handle
Trying to get it moving.
I am not sure the meaning of getting into the oil so but that is of course not good.
With a vacuum operated petcock i the flow stops with stopped engine
But since you don't have that the tank content can run through .
If this was me I would take off the tank and flush it ..getting dirt out
The tap removed ( perhaps a gas filter is needed )
Then I would inspect the tap ,And install perhaps a new gasket
There are kits
Something like that please google better
https://www.ebay.co.uk/itm/362959137048
One can open the float bowls from under with the right tools and
flush and try to clean but I would not do that .Directly ( A bit Red neck Service )
I would then put a clamp or something on the fuel line so I can stop the flow
If the gas runs through.
Starting it and revving up and down --- clamp and open so the float moves inside.
I think there is a good chance it starts moving.
There is of course the risk that you have dirt inside the carburetors and the bike can stop
But I have had virtually no problems with carburetors over the years .
I believe those are fairly forgiving .. I have had leaking Air filters.
But on some bikes there are no filters.
Take it for what it is worth.
Removing carburetors and do a full service is at times needed.
But a typical winter job .Some times the quick fix and getting it running is a better way forward.
It might not run as well but life is full of Compromises.
I don't think I ever have used the res Position. Filling up after shorter distances.