Wool is mostly keratin and cotton is mostly cellulose so your creature's protein and carbohydrate needs may be met in this way. However, fats are an essential component of human diets and those don't occur in any significant quantities in wool or cotton.
It's the little things
The human body needs many minerals that cannot be synthesized. Calcium, iron, selenium, and others are needed on a regular basis or basic metabolic functionality may be impaired. No iron == No oxygen transport. No good. Some vitamins and some a few amino acids can't be synthesized either and must be consumed.
But my monster is special...
Building your monster so it can synthesize all the amino acids and vitamins it needs to function still leaves the problem of minerals and fats.
An interesting possibility is to get the needed minerals from the dyes used in the wool or cotton. Perhaps green dyes contain copper as the primary colorant, so when the monster is low on copper, it will develop a powerful craving for green cloth. Dark red or black cloth contains iron.
Further, dirty/used clothing may be more useful than clean clothes as worn cloth have salts, carbs, or some proteins embedded in them. This makes this monster the literal Sock Monster that causes the disappearance of the odd sock after a load of laundry.
Fats could come from oilskins which contain larger quantities of fats than normal cloth. Also, unprocessed wool has lanolin oil on it though perhaps not in sufficient quantity to satisfy any significant dietary needs.
Getting enough calories may prove expensive because wool is definitely not cheap after weaving. Cotton is cheaper but still not cheap when you need it every day. Unprocessed wool and cotton may be a viable food source.
We want junk food! Junk food! Wool and cotton are boring! I'm not sure of the chemistry of how to do this but your monster could use the long hydrocarbon chains found in plastics as stock for conversion to fats. But! Plastics often have some funky other stuff embedded in the so eating too much of them may cause er, side effects. Definitely something you don't want to eat all the time.
Humans and animals in general don't use hydrocarbons as fuel because there just isn't much of it on Earth's surface. Normal carbs, fats and proteins are much more common so evolution made us to consume those instead of higher energy petroleum products. Life is amazingly flexible in what it can use for fuel. You could include a kind of symbiotic relationship between the monster and some gut bacteria that consume the hydrocarbons in plastics then convert them to lipids. (Heh! Antibiotics make you starve by killing off that gut bacteria.)
Checking on Chemistry SE would be a good place to sanity check conversion of long chain hydrocarbons to lipids.