itsmyturn. - you did what you did for your own reasons. Only you could decide or really know why you did them. We can only try to understand things from your point of view. I would say you are clearly struggling with life without your dh which is totally normal and undertandable.
Perhaps in your grief (and that of your children) you felt life may improve a little if you bring a much wanted pet into the home. I can understand that fully.
As it worked out, it has not helped you through this very tough time, but only made life seem that much harder. That I can totally understand. We all make mistakes.
You have made the decision to re-home your dog and from what you say above, tilly has found a loving new home with people who are there all day, and also a little companion.
I think you have done a brave thing by firstly saying all this on here (which deserves a medal) and secondly by going ahead with the re-homing. It could have been different. An unwanted dog, an unhappy dog and not a very enjoyable life all round.
Don't take too much to heart from what you read on here. You know your reasons why this has happened.
Good luck