Dragons Won’t Give Dragon Souls

Dragon souls can be saved, so don't worry that the effort in killing the dragon is lost.

Please note that you need to discover additional shouts if you have learned/activated all your current words, this way you can use your saved dragon souls on the new words of the new shouts.

Shouts are discovered on the walls of old ruins, and gained by slaying and absorbing the souls of dragons. As more are slain the users ability to learn and use new shouts increases.

— Wikia - Elder Scrolls V: Skyrim - Dragon Shouts

In order to learn new shouts, you need to find all these word walls; you can get a quest to the next:

Although finding word walls can be very challenging try and use shouts in major cities. Whenever you do this you should receive a note from a courier which tells you "a note for your hands only" this note starts a quest to find a source of power and the current location of the next word wall.

— Wikia - Elder Scrolls V: Skyrim - Word Wall (Contains more details on all the locations)

To actually learn a new shout, once you have discovered the word wall and have a dragon soul to use, you need to manually activate it in the shouts menu. Select the shout and press the Activate button. It should be shown on your screen, but I believe the buttons are:

• R on PC

• X on Xbox

• □ on PS3

Because this is an Xbox 360 issue, console commands, the game-engine based code that runs the game and it's scripts, cannot be used. They are for PC only. And, since Skyrim on the 360 is riddled with bugs and glitches, with only some minor in-game workarounds, the best option is to load a previous save.

You have made it clear in your question that you do not have a save previous to this issue. Let it serve as a lesson to make multiple saves over time; it helps in case things go sour.