Blackspot continues to live in dropped leaves and mulch and can re-infect the plant. You want to clear away the dropped leaves. Bag or burn them, don't compost them. You may want to add a systemic insecticide like Bayer's product. I use both systemic and spray to fight blackspot.
Keeping the plants healthy with watering and fertilizing will help. Time and TLC are probably the best options. Once the rose starts sending up healthy new canes, you can probably cut the older canes.
I am not sure if you can completely avoid blackspot in damp climates. Some roses, especially some yellow owes, are very susceptible. Care and fungicides help.
