We finally did something before it was too late, and decided to take down our trees. They had been shedding branches of various sizes for years, and showing less green on top. We took down 3 over the course of a few weeks and found enough rot to justify the decision.
The truth is the trees were a big reason we loved the place and bought it to begin with, and they gave us a lot of shade and greenery. It's like losing a part of our weird family.