Blackwater extract is used to duplicate a Blackwater environment found in the Amazon and other regions of the world. These water conditions are acidic by nature and it is not uncommon for these habitats to have pH as low as 3 or 4, very acidic water and the fish are swimming there without a problem. Bettas are often very happy in a pH of 6.5. Now, what may have caused your fish to be stressed is perhaps a pH crash due to the use of Blackwater extract, which contains peat extract, which is often used to lower pH and soften water in an aquarium. If the pH fluctuation is not too extreme, less than .2, they should be able to cope with the change alright. My planted tanks at times have small pH variations due to my CO2 injection. But if the pH drop is greater, since pH values are exponential, the fish can become stressed.
What to do? I don't want to cause a huge pH shift in you tank as that might be stressful as well. Test your tapwater and determine what the difference is. If the difference is not great, you could do a smallish water change 15% to bring the pH to a more stable level, and don't add Blackwater to the new water. The key is to lower or raise the pH gradually, and not use altering chemicals if at all possible. But if they're happy at 6.5, once they've gotten used to it of course, it's not a bad pH for a betta. And depending on your softness, it may be an ideal spawning pH.
I find that this is a potential risk when using Blackwater extract and IAL. Their benefist to fish are often many, therefore, they are often recommended. They are great products, and I'm not knocking them, I use IAL myself. These products, however, also can alter pH, a property that some people are aware of, but others are not.
My recommendation is to ease up on the Blackwater a bit. It's a great product, just in moderation. I find that with IAL, I am rather conservative with my initial doses and am especially diligent with testing after I add it in. Good luck to you and I hope your fish feel better soon.