My last Lab girl was on Dexamethasone for pain control in her last two years of life. She had invertebral disk disease and significant spinal spondylosis, so basically lots of bad arthritis. It did not mess with her stomach or anything else that I could see. At her age (12.5 to 14.5) pain control was of primary concern.
You are seeing red blood and tarry stool? The first indicates a bleed near the anus. The second might indicate internal bleeding and I believe is the more serious. Unfortunately the only thing I can think is to call up your Vet and book another visit and take in a stool sample. Good luck, keep your old pup comfortable and happy as you can.