It depends on what you are trying to say. "proud" can be translated to many different ways depending on the situation. 1. 어엿한: expressing self-awareness 2. 당당한: similar to above, but much much stronger, thus rarely used 3. 자랑스러운: it's used ONLY when you are proud of someone or significant achievements. It's also an extreme expression that's rarely used. I am proud mother of three children: 저는 어엿한 세 아이의 엄마입니다. "자랑스러운" would be a total nonsense in this case. Beware of bogus translations here non better than peeking at bad dictionaries.

자랑스러워하는 엄마 or 자랑스러워하는 어머니(respectful title of '엄마')

If you mean 'She is my proud mother', it should be '자랑스러운 엄마(=어머니)'. :)