K,
Sibilance is not usually caused by the room. Remember that when recording at 24 bit you do not need to even come close to 0 on the meters... keep your peaks at around -12 to -14 on your DAW meters.
You probably already knew that though...
I am always warning ppl about too much absorption. Basically your problem, as I understand it, is poor mode distribution - which you CAN
NOT fix. You must work with it.
Try moving the mic and singing with your back to the carpeted area. I think you are getting early reflections that create a comb filter in the microphone. This often happens in small booths where a viewing glass is close to the mic. That's why you often see angled glass on the viewing windows of voice-over studios.
I know you don't have any glass there.. but your wall can do the same thing. What mic are you using?? Cardioid pattern, right?
If it's a Cardioid then the wall behind the mic will not affect it that much. Try putting the mic in the 'yellow' area and you sing from where you used to have the mic.
Please give a try to putting the mic in different spots and doing a quick recording. I am confident that you will find a better sounding spot... all without spending a dime.
But you might want to put a panel up on the door and one on the ceiling. But try the mic move first to find a spot that is not boomy. Then you can deal with the reverb.
Let me know how it goes.