About children embarassment: I´ve experienced the same, for years. Probably the only way that this would not happen is when they grow up listening us singing, since they are babies. Once they reach 6 to 10 years, they will feel embarassed if their parents start to sing in public or even at home, no matter how good the singer is.
I started seriously this singing business when I was 46 and they were 18, 16 and 10. My sons used to flee out when I tried to practice a minute. The impression changed significantly when I convinced them to come to some concert, as soloist (classic and jazz/spirituals) or in the choir (classic). Seeing their father applauded by an audience was impressive for them, they accepted the idea, and at least never throw any negative opinion since then. However, they passed the embarassment to me: I do never practice at home. Let some psichologist explain this to me !

Un abrazo - Andrés