Should we replace shingles that are curling and losing the asphalt coating before winter hits?
Replace it now.  Like TODAY.  That's BAD.
With that kind of bad shingle wear, your shingles are likely to create ice dams which will cause water to back up and leak through the roof, causing interior damage, compounding the cost.

In the summer, the roof won't hold the water because it can't freeze, so the curls will only grab a very little water before it drips free.
In the winter, however, a little retained water will freeze, expand, and capture more water, which will freeze, and so forth.