Gender doesn't really have anything to do with it. But risk management does.
I am more risk aversive at the age of 56 than I was at the age of 40 (when I smoked and drank and drove fast cars).
Your two incidents both appear to be a matter of a little overconfidence. Even the second one, you admitted that it was a little dumb to be carrying the overweighted tote bag.
So the answer to your question is simple. No, you are not too old to be riding a bike. People in their 70s and 80s are doing it, and women in their 60s are doing the trans-Am ride like you dream about.
But, there is a definite need to ensure you think about the risks of what you are doing, especially when you are about to engage that is a little unusual (such as carrying an overweight tote bag over your shoulder on the bike). There may well be alternatives that mitigate the risk.