Yep, you need a new exhaust hose. You should also expect to need a new water shutter, which is easy to replace since you'll have the exhaust apart.
If your shutters are burned, make sure you remove both metal plates. If the shutters melt the metal plates will slide down into the exhaust. This won't hurt anything unless the shutter blocks the exhaust.
Check condition of exhaust bellows also.
2006 Bayliner BR205
A1 Gen 2 1.47
5.0 carb engine