Is it correct to write "All of the people had a bag" if each one had a bag of their own (and not one shared bag)or "All of the people had bags."
I realize that it would be best to wirte "Each person had a bag." But if the sentence begins with "All of the people had..." should it be continued with "a bag" or "bags"?

If you mean that they had only one bag each, you would say so:
"All of the people had a bag."

If they had more than one bag, you would say so:
"All of the people had bags."

Of course the fully correct sentence would be:
'All the people had a bag EACH' or
'All the people each had a bag' or
'All the people had ONE bag Each'