Yes, your entire muffler needs replacing. There is no way to get inside and weld the baffle and if someone could it would cost more than the new muffler.
Your service light probably would come on if the cat was plugged because your exhaust would not be getting cleaned as well and the O2 sensor downstream of the cat would send a signal that it has high hydrocarbons. You can go to most auto parts stores and have them scan for stored codes. Sometimes it will store a code and fail to turn on the light.
If you do a lot of short trips where the engine barely gets warmed up it will rot your exhaust system faster and contaminate engine oil and tranny fluids faster. Avoid short trips if possible.