Welcome to RS Ellysa!
No matter what board you post this question, you are going to get the same responses. Whoever told you that a carpet in a 12 gallon would be ok needs to crunched in one himself.
To understand why, you must understand how the anemone functions. These things use the water like their blood. They take in nutrients and they expell nutrients through the water that they live in. Like all animals, they excrete. In a 12-gallon system, they are regularly ingesting their own waste. Who would like that?
Of course, you could counter with daily water changes, but the problem with that is that this animal requires extreme stability. Even minor fluctuations in the salinity of your water changes can unbalance the water.
Another problem is lighting. Anemones require high lighting, and unless you are perhaps lighting the system with high grade LEDs, it is almost impossible to keep enough light over the system w/o causing temperature spikes. Turn the light off at night, and 10-12 gallons of water can drop 4-5 degrees quickly. Again, there goes your stability.
Anemones also need to be fed. In a 12 gallon system, water can foul so quickly. IMO, systems this small shouldn't be fed at all, and if so, very sparingly. Not gonna work with a nem.
Alas, in your situation the best way to save your carpet is to return it immediately. As mentioned in an earlier thread, you are probably already doing damage which may preclude this animal from a healthy recovery even in an appropriate tank. These animals can't show distress like many other animals. They look great one day, and dead the next. Sick anemones don't stay sick long.
Finally, keep in mind that these animals regularly fail in very large systems as well. Sometimes it has little to do with your reef-keeping skills at all. They ship very poorly, and more often than not fail in even the best of captive environments.