I have a very busy home as well - I normally have free time during they day as I work from home but the evenings it's dinner, homework, baths, wife's honey do this honey do that..
I play a necromancer - so I'm not group dependent. I can kill a mob or 5, afk for 30 minutes and no one misses me.
So my suggestion would be to play a very soloable class and when you can contribute to groups great, but when you can't you can still get some you time in between family needs =)
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