Greg , DNA doesn’t lie - you have presented the information and the return replies reflect your situation . Perhaps for real piece of mind you make contact with the female ( half sibling ) and both do DNA samples within your GPS or pathology departments .. best of luck.
If you are sharing the 1537cM with this female It would be half sibling based on both your ages. Share the same father , different mother . You both share the same Paternal Uncle . Based on the information you have provided.