Oh I am so sorry, both that you lost your well-loved Terrier, and that your Dalmation has taken it so badly. Grief and missing a loved companion is natural in dogs, but I can understand why you are worried, when it has this effect.
Yes, depression can cause physical decline, but even an old dog who has got sad and down, that she feels her age, and has aches and pains, can maybe be cheered up.
I don't really know about how wise it would be to get another dog in.