Sounds to me like swim bladder disease. I have a dwarf powderblue Gourami that has it right now as well. She has a slight bulge in her side, but mainly either swims on her side or even upside down. Hasn't eaten in over 6 weeks.
The swim bladder is the organ that assists the fish in moving up and down in the water. Gourami also regularly take breathes of air from the top of the tank and ones with swim bladder issues can rarely do this.
The typical and EASY way for most people to cure this is to cook some peas, then peel them and feed them to her (it's great if the other fish like them as well ) If you've ever purchased some garlic extract you can soak them in a little of that and then they''ll really be a big hit. The fish is basically constipated (usually due to all the dry food we feed them) one they poop then the swim bladder disease usually corrects itself but I would still feed peas periodically - peeling them is a minor pain, they usually just slip out of their little jacket and it will reduce the incidences of fish bladder.
Mine is still not well primarily because she refuses to eat the peas so I tried mushed up peas in water added garlic extract and a little bit of Miralax (a mild human laxative), and feed them through a syringe (no needle) but she still refuses food for the most part. Her swim bladder issue has come and gone multiple times and she allows me some liberty to feed her by hand. She too will try tubiflish worms diet but often burps them out. Originally she wasn't coming up for air at all. Now she does usually try to come up for air. She does not seek out food but only eats when I direct the food towards her. After this long I'm afraid she has some death to her intestinal tract but she still keeps fighting so I haven't euthanized her yet.