No crawl space.If you have 0" between the sill and the dirt then how do you have any crawlspace?
No.Slab jack. Apparently the subgrade is quite firm and you do not need footers. Can you just pump grout into the area between the floor and the dirt to seal up the leakage? The beams will be unnecessary if the entire floor becomes a slab on grade setup.
I do need to remove the cabinets as the wood floor goes under the cabinets. This is not your typical Home Depot bought wood flooring. This wood flooring is about 200 years old, very wide, very thick, and very hard. The cabinetry would need to be removed.You do not need to remove cabinets to appy a new floor with proper air seal. Of course it would be better but the cabinets themselves should be sealed fairly well.
Again, there is no crawl space. There is no concrete foundation. There is no sill as you are imagining it.Lastly, You have concluded that no work can be done to prevent air from passing through the floor. That air got under your house from the perimeter somehow. Seal that leak at the perimeter. Unvent your crawlspace. brick and mortar around the perimeter to a height above the sill.
There is no sill plate. There are massive, hand carved beams that act as structural support with some sort of stone foundation at the corners.I am most confused about how you state that the sill plate is on the dirt but somehow air is coming up through the floor.