I wouldn't go to a pro studio to record your sax unless A) you have money to burn, B) You have NO microphones or preamps (you DO have a mixer, don't you?), C) You never used recording software in your life (whatever you are reading this with is powerful enough to record a sax onto a backing track), and D) you are going to be comfortable, unhurried and confident once in the real studio, with all it's intimidating gear. and engineers looking at you.

Probably better to put a small recording setup together using your computer and just experiment until you get a decent sound. It's not rocket science. Just record when the house is quiet (although a sax is so loud, background noise needs to be pretty loud to get onto the recording!), and listen carefully to the recording to make sure reflections and room reverb don't get too loud.

Then a simple, soft LA2A-type compression is added (in software) and you are done! Now you can record any time you feel like it, no extra cost..!

Sure, if you are recording for Sony Records, they'll expect you to do it in the studio. But I have heard a TON of well-produced home recordings. Just don't expect the best results the first time you try, experiment until it sounds good (hire an engineer to come to your house and set up the stuff for you if you are a total newbie).

Better to learn to fish and cook than have to go to a restaurant EVERY time you feel like some grouper!