Try not to be too hard on yourself.
I know that you feel that as a parent, it is for you to be 'strong' for your children, but there is nothing wrong in your children watching you grieve as well. It helps them to understand that everyone feels the same way about your dad, and that his pasing is going to take time for everyone to adjust to.
This sort of event - where something will spark of a memory, and bring out the attendent grief, is something that will happen again and again. You should never fight it - give in to your feelings, and all learn to comfort each other.
We British are notoriously bad at grieving - we act as thoguh a month on, everything is back to normal, but it simply doesn;t work that way.
In time you will all be able to talk about your dad without the pain and tears, but while they are htere, let them through, it's good for all of you, it helps and it heals, so don;t fight to control yourself because you feel you have to be the 'strong' one.
You will support your family, but don't push away the notion that they can also support you.