First, I want to let you know, that I only had a sax player in my home studio once. So what I write comes from research.
If you use a large diaphragm mic for sax you tend to go away from the instrument. Some like placing the mic over the shoulder, others make a triangle with mouthpiece and bell. So you get to record a lot of room. Your bed room might actually sound good, especially if it has lots of filled book shelfs in it.
I had to treat my recording room a lot. If I clapped, I got a lot of very fast reflections, which gave the sound a metal feeling (difficult to describe).
If you have a carpet, it will probably, together with the mattress, steal some high frequencies.
But maybe that's just what you look for in you sax sound.
As I found out, a lot of known sax players use SM58 and some SM57. Some even for recording. Maybe you should try to find you SM57. 😉
The room is not so important in that case, because you won't go halve a meter away from the mic.
You can also think about a small diaphragm condenser like e.g. the Oktava MK-012 or something similar so you have a mic to record guitar.