The problem is probably to large of a batch... never more than 6 cups of the prepared fruit. After that, not enough cooking. The fruit/sugar mixture needs to be boiled gently until it reaches the "sheeting" stage. Keep a large metal spoon in a cup of ice water... when the spoon dipped in the mixture "sheets" of the spoon rather than runs it's ready.
Additionally, as already suggested, pectin can fix this problem... even now. Place all of the mixture (should have been one cup of sugar to one cup of fruit) in a heavy pan and bring to a boil. Mix 2 packages of Sure Jell Certo Pectin (powdered) into 1/2 extra cup of sugar and stir into boiling mixture. Boil hard for one minute and the place in hot jars, Place lids on jars finger tigth and put in yet another pan of hot water that covers the jars just above the lids and boil for 10 minutes... remove and let cool.
Under normal conditions use only 1 and 1/2 packages of the pectin.
Your proposed blackberry jam (we call it "preserves" here in the western U.S.) should be canned the same way.... squash the fruit somewhat with a potato masher to bring out the flavor (be sure to use new jar lids, don't reuse the old ones!)