Singing <correctly not professionally> and being able to improvise is absolutely linked IMHO.
Being a multi-instrumentalist here's my advice:
What has helped me and many others is the following :
Whenever I listen to something I like I try to reproduce it. This means I transcribe it, I try to memorize it by humming it with my inner voice, and when I'm sure I got it in my head to play it in my instrument.
Then I try to reproduce it as much close as possible. Including articulation, type of sound etc.
This way I improved my understanding of melodic patterns and I think it's the hidden 50% of how to analyze music.
For example what sense does it make to know how to use 11 or 9 chords if I can't "hear" that 9 or 11?
Why has this artist used these chords in this song I liked, and how does he play his sax over them?
I discovered these things by studying my theory ( 50% ) and by seeing who uses that and why by transcribing songs ( other 50% ).
And why am I saying all this? Because this apparently has made me a better singer. 10-15 years ago I was really really false. The time between thought and actual singing was leading to really really bad singing.
Lately I can hum the correct root note or a whole melody of a song without listening an other one as a reference.
And people tell me that whenever I sing a tune playing the piano or guitar that I could well be a singer. ( Of course I cant. But using this technique I'm much better in achieving the correct pitch because I hear the correct one in my head ).
This has helped me an AWFUL lot in sax playing. I'm playing for almost a year and it was a lifesaver with my embouchure exercises, achieving really quick to play overtones, and to solo effortlessly in a band without "preparing" for a solo.
It saved me some money too. With my single mouthpiece/ligature combination, I'm trying to imitate as much as I can the feeling of sax players I like and I discovered this method makes me a lot more versatile than a good friend of mine that has a drawer full of Mpcs and keeps exchanging them. He's quite good. But not practical.
Sometimes when I improvise a solo I hear my inner voice singing a melody that I don't really analyse and play it, sometimes I think <hmmm ..... it would be cool to play some diminshed lines and break it with a minor blues scale bar etc.... >
So, summarizing,
listen to a lot of good music.
Spot a passage that is a little bit difficult for your level but you would really like to play.
Listen to it again and again. Let it get into your head.
Try to hum it correctly!!!! <recording your voice can help align those faulty notes there mate!>
Once you have it in your head looping, try to play it. <Isn't this the most fun part in music ?>
After the first 15-20 tunes tell me if this hasn't accelerated your skills or not
( In sax, singing or any other instrument ...)
Cheers.
Stelios