How can I convince my family to wear the legitimate head 'abaya after they rejected it on the grounds that its appearance is unacceptable, and is it permissible to buy it without their knowledge?

The legal hijab for a woman is to cover her entire body from non-mahram men with clothes that do not resemble men's clothing and do not describe or define the limbs. There is no specific type of clothing required. If these conditions are met in any abaya, whether it is a shoulder abaya or otherwise, then it is a legal hijab.

If the shoulder abaya is modest and fulfills the conditions, then there is no blame upon you for wearing it. However, if it is not modest, then it is not permissible to suffice with it, and you must cover the rest of your body, as a sixteen-year-old girl is mature and must observe complete modesty. No attention should be paid to customs or traditions that contradict this.

Whoever seeks Allah's pleasure by incurring people's displeasure, Allah will be pleased with him and make people pleased with him. And whoever seeks people's pleasure by incurring Allah's displeasure, Allah will be displeased with him and make people displeased with him.

Buying a head abaya with your own money is permissible, but with their money, it is not permissible except with their permission. You should deal with your family wisely and gently to explain the truth.