Slate roof

[tcfjen]

I've been browsing the forums for awhile, but now it's time to get down to business with my family's new-old house. It's a mid-atlantic farm house (folk victorian) probably built somewhere between 1880 and 1910. We have a slate roof that is most likely covered in Peach BOttom slate which is around 120 years old. As far as we can tell, the slate look good and no major leaks. I'm sure there are flashing issues, but that's not the focus of my question...

When it snows we are getting piles of snow throughout the attic from snow that filters its way in. There are no large holes or areas of daylight showing between the exposed slates, but somehow the snow works its way in. I've had other people in our area say the same thing happens to them, but no one has a solution.

I would like to finish the attic with insulation, etc and make it a living space. The entire house also lacks hvac and that too is planned to change. But I don't want to make these changes and have it snowing inside the house.

Any solutions? Or do I scrap an otherwise functioning roof that could have another 80 years of life left?
Thanks for any input!!
Jen