I first had CBT and hated it - I think it was the therapist I had, and the fact that it felt rushed and just not what I needed at that point.
I have been seeing a therapist privately, for 2 years now, and she has used a mixture of things, and to my great surprise, she has also used CBT successfully with me. I have been depression free for 2 years now, and have even managed to handle a miscarriage recently.
What I'm trying to say, is that I think finding the right therapist is the most important part of getting better. I have had short blocks provided by the NHS before, and they have been a sticking plaster - not working for long.
If you can afford to find one privately - I pay £25 for a session, and my therapist will negotiate a lesser rate for those who would struggle to pay that, as I'm sure others would.
Maybe go and see the one the doctor has sorted, just to test the water. You'll know for definate then if you can't go through them.
Good luck - you can get better with the right treatment, and devoting some time to it.
Ps. you are not a failure - you have a mental illness that is bloody hard work!