First thing, you should use an enzymatic spray to clean up the messes on the carpet. You can buy these at any pet store, and they will do the best job of cleaning up feces and urine on the carpet. Cats have really sensitive noses, and they are able to smell the remnants of a mess on the floor even when we can't. If they can smell it, they may think it is an okay place to potty.
Ruling out a medical reason for the source of your issues is extremely important, like I said. If cats experience pain when they are going to the bathroom, they can start to associate the litterbox with the pain. This causes them to not want to use the litterbox due to not wanting to feel the pain they experience when they are inside of it. Things like intestinal parasites, inflammatory bowel disease, anal sac impactions, and gas can all cause cats pain during defecation. Resolving the pain issue will similarly resolve the potty issue.
Behaviorally, the best way to eliminate the out-of-box eliminating is through helping to reduce the anxiety. (Which can take some trial and error, trying to discover what it is that put her off of it in the first place).
If you've got a two-cat household, your covered litterbox may be the issue. Sometimes one cat will stalk the other cat while s/he is inside, by sitting on top of the litterbox and pouncing or attacking the other cat when they come out. This can be a very negative experience for the cat inside! If your cat has been scared by the other cat in the household, he or she might not want to go inside. You can try taking the cover off the litterbox and see if this helps the cause of the issue.
Similarly, loud noises or sudden movements from outside can startle and alarm a cat inside a litterbox. If they've had a frightening experience while pooping, they may avoid the use of the litterbox to avoid the source of the fear. In these cases, you can try moving the offensive litterbox to a different location to try and entice the cat to use it again. Sometimes providing multiple litter boxes throughout the house in several rooms will help to get them started back into the litter box, and allow you to see which ones she seems to favor. IE, she likes to go in the litter box in the dark corner of the bedroom, but the one next to the loud washing machine is avoided completely. Once she is more resolutely going in the boxes instead of in the furniture, you can start removing the extras that she's not using.
If you've changed litters recently, sometimes the smell of the litter can be the cause of one cat not using the litterbox. Cats may not like the fragrance of the scented litters, or simply the difference from normality - cats are very resistant to change. Sometimes switching back to and old, or trying out a new, unscented litter can help solve the issue.
Clean the box frequently. Some cats are extremely picky about digging in a litter box that has urine and feces inside. You may have to increase the frequency that you scoop the litter. If you do it once a day, try doing it twice or even three times. Some cats just prefer a REALLY clean box.
Positive reinforcement is always better than negative reinforcement. Some cats will respond to positive reinforcement when using a litter box. If you've tried moving, changing litters, plug-ins, extra boxes, and are still seeing no success, you may have to try re-training your cat to use the litterbox. The best way to do this is through praise. If your kitty likes treats, praise her and give her treats every time you catch her leaving the litter box. If you see her squatting to pee, pick her up and place her inside the box. Again, treat and praise when she exits the box. Don't punish or raise your voice to your kitty, or you might cause her more anxiety which she may associate with going potty. This is really time-intensive, but sometimes when they start outside the box, it's a bit of effort to get them back in.
Isolation can take away the other appealing object which might interest her. She may have simply learned that she likes to potty on soft things. You may need to isolate her in a room with food, water, stimulating toys, and her litter box away from the tempting objects in the living room to encourage her to use the litterbox more frequently. Usually after a couple of weeks, they have become in the habit of using their litter box again, and are safe to roam free in the house.
Finally, you can try using a pheromone spray or plug-in in the rooms with the litterbox to help reduce anxiety. These sprays and plug-ins are designed to release chemical scents (undetectable to us) that reduce out-of-box habits, and produce a calming, relaxing environment for the cat. The plug-ins are just like a glade plug-in, and many of these products can be purchased at your local vet store. The most popular product on the market is called "Feliway". I suggest putting them next to the boxes themselves, and again in the room where she is doing a lot of pooping to help reduce the stress in those particular areas.