It does not help to be told that if I was thinking right, then I wouldn't care how I was feeling.

It might if someone said it right. I had similar unproductive internal monologues, and unproductive feelings as well. But eventually I determined that the feelings themselves weren't the problem, it was the "I'm defective" death spiral I engaged in when I felt those feelings that was the real problem. Why do I feel bad? Why can't I do this? What's wrong with me? Blah blah blah blah blah. The feelings weren't hurting me (much), my reaction to them was (a lot). The reaction I've promoted, and which has helped, is "It's just a feeling. It sucks, but it won't kill me."