Hi Mama-
First things first, you need to take puppy to the vet. I know she is old and the doctor will probably say something like she has cancer or she is pretty much at the end of her life. If she is suffering, which all the symptoms point to her being sick and not feeling well, if you cannot afford $1000 for a few more months of life, then allow your children and your family to say goodbye to cricket and have the dog put down. 16 is a good long life for a doggie.
For your kids, have a talk with them about life and death and about (I know it sounds corny) but the circle of life. The body that we have (christianity) is only a temporary shell for the spirit that lives within us. Some people say that animals are exempt, but I tend to disagree, believing whole heartedly that everything is connected and we all do the best we can here for the creator we depend on until we can be wholly in the presence of heaven. Your puppy just beat you to it, that's all.
I do know that there are books for kids about death, especially regarding grandparents, and I am pretty sure there are ones about pets, too. Either way, please tell your children that you enjoyed cricket's life and that she had a good one. And, when and if they are ready to take on a new pet, you can go to animal rescue together and pick out another one. Reassure them that no one can ever replace cricket, but that also it is cricket's time. It's okay for them to love her and to be sad, and it's okay for them to not want a new pet because of the process of life and death. However, if they feel that way, then that needs to be addressed so that the sadness and fear of death is not permanent, but to celebrate the life that was and to also celebrate the new lives that are still to be.
Good luck Mama
-E. M