What conjugation rules are applied in 恥ずかしながら?
As Zhen Lin explained, 恥ずかしながら is formed by attaching the particle ながら (which means something close to “although”) to 恥ずかし, which is the stem of the i-adjective 恥ずかしい. However, this is not the regular pattern of ながら in modern Japanese.
The regular pattern is that when ながら is attached to an i-adjective, the adjective takes the dictionary form. Web search shows many examples of 美しいながら, 弱いながら, and so on.
So if the phrase followed the regular pattern, it would be 恥ずかしいながら. This form is not unseen but less common than 恥ずかしながら. I do not know why it is. It might be the case that the two vowels ii were contracted to one vowel i because the phrase is used very often.
Daijisen (sense ) states that this ながら was attached to the stem of an i-adjective in old time. However, I do not know how “old” this “old time” refers to, and I do not know if it explains the reason why 恥ずかしながら is more common.