Why does my GPU run slower after a while, and is it damaged?

Almost always this is thermal throttling, not damage. When a GPU gets too hot, it automatically reduces its clock speed to protect itself, which shows up as a drop in mining performance after the card has warmed up for a while. The fix is to improve cooling rather than to worry about the hardware. Check that case airflow is adequate, that fans are clean and spinning, and that the card is not starved of intake air. On a used card, dried thermal paste and worn memory pads are common culprits and respond well to a repaste and repad. Setting a power limit of 70 to 80 percent also reduces heat substantially and often eliminates throttling outright while barely affecting hashrate. Sustained throttling is a cooling problem to solve, not a sign your GPU is failing.