If your ability to hear and accurately reproduce music is indeed tied to some sort of neurological impairment or developmental delay, there are all manner of ways to wake up those underused synapses and get your brain/ears/voicebox working together more reliably.

You have the right instincts already: picking up instruments and working on learning tunes, working with a teacher, trying to teach yourself songs at your own pace, training your ears, etc. I'd suggest you seek out additional ways to fully embody the music, so you're learning with as much sensory input as possible. Lean your body against a radio or piano to feel the vibrations. Try to clap or tap your foot or march around the room to the beat. Rest your hand gently on the front of your neck when singing and experiment with different levels of relaxation and tension. So what happens when you try to "move the sound around" in your body. Also play with the sounds you produce using different kinds of breathing.

You might even try to get your hands on a book or website that describes "sensory integration" or "brain gym" techniques. While these exercises are designed primarily for schoolchildren with neurological impairments, they can prove helpful to people at any age, whether or not you have a diagnosed impairment.

Best of luck!

--Cuilionn