I grew up in a musical family. My dad always sang and taught me songs at a very young age. My mom, on the other hand, probably never sang a note in key in her whole life. However, I loved to hear my mom sing. She made up her own words to familiar melodies and made it really fun. Dad was more serious about it and, unfortunately, ridiculed her constantly. As a result, she never sang when he was around. I remember feeling saddened by that. It was much more fun when we all sang together as a family, but he made it too hard for her to participate. Thank goodness he didn't stop her altogether, though.
It is so important to model the joy and importance of sharing music together. Children need that experience. The only way to do that is by jumping right in and doing it. That means that everyone gets to join in, regardless of their abilities. However, if you want to improve that ability and learn to sing closer to "in key," there is something that will help. You can practice “audiation.” Audiation is the ability to hear the music in your head when it’s not actually playing. When we get those ear-worms that just won’t leave, we are audiating.
So how do we practice this? It's simple. Try to hear songs in your head - audiate. If this is difficult, sing a simple song all the way through. Then, leave out some of the lines in the song, following along in your head. If this is a real struggle for you, it may take a little while, but keep at it. Just like anything else, this will get easier with practice. Try to look at it as a game. You can do it when you're alone in the car or when you're singing in the shower. And, you can just leave out a word here and there, eventually building up to omitting whole lines. Most importantly, don't give up. Your children will thank you for it. Good luck!