(Excerpt from a discussion about using sound as a healing modality.)

[...] holding the intent to get the perfect teacher is also a key here. My mom was completely tone deaf. Like COMPLETELY tone deaf. She called several music school, and also numbers for voice teachers listed at the local store. She told them exactly what she wanted, her problem, and discussed if they could help her. Some believed tone deafness was incurable. But she found a teacher that not only helped her and coached her out of it, but was super strict about her hours of weekly practice (which she also needed because she was very lazy). He did not believe in tone deafness.
To tell you the truth, having grown up with my mothers attempt at singing (she never stopped singing even though she was awful at it), I did not believe the teacher would have any effect. But after six months, mom said she wanted to sing a song for us, we all sat down dreading what was coming, and OMG the most beautiful voice and song came out of her body. I was shocked. After that, I never underestimated the power of transformation.

My suggestion would be to shop around for a voice teacher and explain to them what you want to do and why and what your barriers are, including the emotions that come up when you don't get it right. The right teacher will know exactly how to coach you into developing this skill. And the wrong ones will tell you it's impossible :)