Most of the plant looks well but there are a few branches that are completely dead- I just don’t want to be missing something and the whole thing dies. 

I think your plants are infected with a peony fungus that is causing the plant to darken and rot.  There are two fungi that may be the culprit, Phytophthora Blight (Phytophora cactorum)or Botrytis Blight aka Grey Mold or Peony Wilt (Botrytis paeonia plus several other species).

Phytophthora Blight of peony  is a soil borne fungus that thrives in cool wet soils causing young shoots to blacken and rot at soil level. Roots may also rot. Higher up on the plant leaves may develop black leathery areas as well as flowers and buds. Symptoms of Botrytis blight look similar to Phytophthora Blight, but it has the presence of a fuzzy grey mold. I see what may be some of the mold in one of your pictures.

Managing both problems is about the same.   Remove the  infected plant tissue or entire plants and soil when most of the plant is infected. You seem to have only small plants infected.  The group in the back of the photo seem to be healthy. Discard infected plants. Clean the soil carefully.  Do not compost any debris. Good sanitation is critical in managing and preventing  the spread of the disease. Be sure the soil drains  and that there is good air circulation around plants. To prevent the probem next year, begin applying a protective copper-based fungicide early in the growing season. Avoid overhead watering.

The following websites may be helpful. I hope you are able to control these problems, especially since these plants have been in your family for so long.