its not so much the carbon, but the bacteria that are important. Basically what happens in a tank is beneficial bacteria build up to break down your fish's waste. The majority of these live in the filter, which is why many filters have a spot for the bacteria specifically, like a spounge. This spounge should NEVER be replaced, only rinsed off with dechlorinated water if it gets too gunky for water to flow through. Right now, you should have some bacteria in your tank on gravel and decorations, which will help. If your filter doesn't have this, look around and try to find something you can place inside the filter for the bacteria to grow on so you can change the carbon without going through this every time.
For the time being, get an ammonia test kit tomorrow, so you can keep an eye on how bad your cycle is. Also keep up with daily water changes. If you need to do several durring the day, go for it, sine that will be les stressful than the ammonia.
I have no clue what it could be if the fish doesn't look like what is in the pix, those were my two guesses. But clean water will definately help them, so keep that up.