Honestly, carbon isn't necessary to good filtration but it helps when trying to remove chemicals from the water eg. meds etc

The best filtration for goldies is both biological and mechanical so sponge and bio balls (ceramic noodles etc) to house good bacteria...
If you can, it would be helpful to see what your ammonia, nitrite and nitrate levels as Laurie asked as it's important to rule out water quality issues. Bacterial infections can take hold when fish become stressed and one of the biggest stressors is water quality. It's also important that if meds are needed, water quality is excellent because medicating with toxic water can do more damage than good.
So if you can, please post back some numbers on:
- ammonia
- nitrites
- nitrates
- pH
to paint a clearer picture of what's happening. If you don't have test kits, perhaps you have a local aquarium that can test the water for you