So, essentially what you mean is, when do we change the tense of a verb in reported speech and when do we not?
Yes, both statements have the proper grammar. Furthermore, there is no advantage to choosing one over the other without context.
The two are only slightly different from one another, so you can typically get away with using either one without raising any questions. The context, though, is crucial. Both of your generalizations are smart choices. I'll include a third: Is the circumstance still pertinent?
"I was recently stopped by two people who wanted to give me $1 to board the bus. They said they had no money." Maybe I just experienced this five minutes ago; perhaps the panhandlers are still out there. However, as the person sharing a personal experience, the occurrence is firmly in the past for me, therefore "didn't" is a preferable option.
"I am aware that my grandparents still owe the debt. However, they are unable to pay it at the moment. They said they had no money." Even if the conversation about the lack of money happened two weeks ago, it's still relevant to the current conversation. It is implied that the grandparents are still in need of financial assistance.
Thanks,
Eddie