Ridge vent - no soffits / no ability to install

Hi all

I think my GC talked me into something dumb. Last year we had some water coming through the attic. A GC came out and they took down the chimney that was causing the leak. He also recommended installing a ridge vent - saying it would vent the house. Turns out I don't have working soffits, and literally my house has 6" overhang so like putting soffits is all but impossible.

So now I think I have ridge vents.... But nothing it's able to vent it? I mean I have a few roof vents but I don't even think they go all the way through the house.

Any idea of what to do next? I think I'm getting very minor ice damming in one part of the house......... I hate houses 😅