Thank you so much for the advice! That's kind of what I was thinking on the piping.
My husband suggested using his dremel with the wire brush attachment to remove the interior crud. Some of it we think is actually glue! I'm going to have to look at it a little closer before I decide what to do.
Here are a few pictures to illustrate. I also have a thread in the Dooney forum about this, but there doesn't seem to be as much vintage bag chatter over there.
This is the bag overall. I love it and love the size, otherwise I would not mess with this much effort.
This is the interior:
The damaged piping corner: