I have the same question and the story is undoubtedly the same. I have additional details of the story but do not know the author or title. It is not "The Survivor's" as suggested in the answer.
Earth is in desperate ecological shape. In the first chapter, scientists discover that no matter what humanity does, the Earth will become uninhabitable. So humanity begins a crash course to develop interstellar drives and spaceships to move humanity.
Our first test of the interstellar drive creates a black hole in the Alpha Centauri system. This disrupts the orbits of the planets in that solar system - much to the detriment of the intelligent species living in that solar system. They are not a technological civilization but a psychic civilization. They send out a telepathic distress message identifying humanity as the race guilty of their imminent genocide.
Most of the rest of the book tells of how the psychic races of the galaxy attempt to take retribution on humanity while humanity attempts to flee from persecution by the galactic telepaths - this includes setting booby trap ships and planets.
The book is somewhat written as a series of stories mapping the progress of humanity over time and how this hostile environment forces humanity to adapt in order to survive.
Human ships are dumbbell shaped. Late in the book, humans paint the outside of their ships with scenes of Earth as a memory of what they lost.
Late in the book there's a mentally picturesque meeting of humanity high above the plane of the galactic spiral arms.
For a while I thought the title was something like "Earthsong, Star ...?" but I can't find any novels similar to this. It is not "The Songs of Distant Earth" by AC Clarke, "Earthsong" by Victor Kelleher, nor is it "The Dispossessed" by Ursala Le Guin.
The book was likely published before 1990.