I chose band because I can't sing...?

I chose to do concert band (I play saxophone now but I have experience
with oboe) because the other option was choir. I choose not to sing
(unless I'm in my dorm and youtube is blasting), because I'm
dreadfully out of tune. Sure, I sang in elementary school and kid's
church choir, but I didn't care if I sucked then.

I've discovered in college, that band and choir are connected somehow.
My professor said, "If you can sing it, you can play it." When we warm
up with tuning notes, we have to sing the note, and then play the
note. I can play the note, but I refuse to sing.
Sometimes, he goes around the room and makes us match pitch with
singing. When he pointed to me, I got nervous and refused to open my
mouth. I had to do it, and my singing voice was flat and probably
painful to listen to, and I was embarrassed.
I hear girls describe their singing range, and based off of my
experience, it feels like if you're a girl, you have to sing soprano,
and soprano is high pitched. My voice is deeper in comparison to other
girls I've met, so I have no idea what my range is.

Question #1: How can I get over my embarrassment when it comes to
singing? I've never had the aspirations to be a famous singer, but I
do have a little bit of a performance anxiety with the saxophone.

Question #2: Once I get over this embarrassment with the singing, is
there a way to determine your vocal range? I know I'm definitely NOT a
soprano, maybe somewhere near the middle to low range?
to think about vocal range, I think about what I am in terms of
musical instruments. I definitely not a piccollo (soprano), but I'm
not a tuba (bass). There's female ranges and male ranges, is it
possible for a female to sing with a male range? Or is that determined
from the structure of your vocal chords and it's different based on
gender?