CBT can work at different levels; you can learn how to notice the NATs that arise to the surface then challenge them. So you're not necessarily stopping them but addressing them once they arrive IYSWIM. You can get to a point where you notice the NAT but don't necessarily believe it, hence the CBT mantra, thoughts are not facts
Sometimes working with the NATS is enough for some people.
Or with a bit of digging around with your therapist, you can work on your underlying core beliefs about yourself, people and the world. This is working at a much deeper level of thought IYSWIM but it is possible to make changes with some time and effort.
If you've been thinking in a certain way for aaaages, it's not going to turn around quickly. It takes practice and thought and time.
I'd also recommend Headspace as pp said above. It really helps with the noticing of thoughts and letting them pass with self-compassion and without judgement.
Mood gym wan't quite right for you, but there might be other stuff out there that's more relevant.
Feel free to PM me if you want anymore info 