Wait. You knew your betta was being shocked by each water change and did nothing to correct this? You should have expected far worse long ago. This sounds to me like a bacterial infestation, something along the lines of septicemia. Is blood leaking into the fins or actually clouding the water around them?
With the approaching senility of your betta, frailty will ensue and you will need to be changing his water more gently from now on or risk worse and worse reactions, okay?
Is it the temperature or the pH which is different in the two waters? Something is freaking your guy out of he's laying on the bottom every change. Healthy transitions typically involve exploratory behavior and fiesty resistance, not any kind of sulking or sinking. How long does he lay on the bottom exactly? More than ten or fifteen seconds?
PH test kits are usually super cheap if you don't have one. Liquid is better. I think the strip type is sprayed on or dipped on, both methods will leave variables in test accuracy.
If nothing else, treat him like a new fish every time you change him, gradually acclimating over the course of about an hour. This will greatly ease the transition. It is better, if you don't have a thermometer and are a bad guesser, to move a fish into *slightly* warmer water than cooler. By slightly, I mean two or three degrees, just enough to feel by swishing your hand in the new water. I'm not sure what could be stressing him if you let his new water acclimate to the similar conditions as his old ones. Please tell more about what else might change the water in either. Any shells or other dissolving decoration in the water? We can figure this out!
For treatment, since water changes are seemingly part of the problem, I'd advise using a broad spectrum antibiotic at appropriate dosing for his tank, but cut it in half so that you are only changing the water every other day.
Any other opinions to help our friend?
*there is the sad, slim chance that he is ready to move on into the next world simply by virtue of age*