The rules of capitalization in the 18th century were much more permissive; you'll note that she also capitalized Heart, Dearest Friend, Friendship, Letters, and Hands. Many nouns were capitalized, and even some of the adjectives attached to them. This practice died out over the next century or so, leaving us with only a few things to capitalize regularly.
As to the turn of phrase, yes, I think you have at least the basic gist of it. Abigail is not able to find out how many letters have actually been written vs. how many have been delivered, she only knows that only a few letters have made it to her so far. But she does not believe that her dearest friend has not actually written to her often (because it would be unjust to assume that this friend doesn't care enough to write frequently); she can only assume that it is because of misfortune that she has received only a couple of them so far, and the rest are still in transit (since mail service was also quite slow and irregular in those days, especially over longer distances).
The phrasing of "sit down to the score of misfortune that ..." is similar to the more current "put it down to X (that Y)", meaning that you believe X is the cause of Y. After your friend has done something bad, you might say, for instance, "I can't believe that John did that to me on purpose; I've gotta put it down to ignorance and bad timing."