I know what you mean about that vibrating palate! Sometimes my teeth vibrate so much I fear I'll lose dental work. Then the palate itches, and it feels as though flesh will separate from facial bones.

It sounds as though singing "ee" is producing a harmonic resonance in the bones of your head. The only thing to do is to change the sound you are producing so that the vibration doesn't jive with that of your bones.

Knowing what region you are from would help me describe vowel sounds more clearly, but try these changes:

Sing a more open, less nasal sound by raising your palate while singing (like when you are yawning, or when a bite of pizza scalds your mouth).

or

Change the vowel sound slightly. Try substituting a short American i sound for the ee. Or add more long a. Usually people don't notice the difference in context.

or

Rewrite the song to eliminate the offensive vowel sound.