It really sounds like the issue is with your job environment. The problem with staying in a job like this, as you are finding out, is that you get so discouraged that you start to think there's something really wrong with you-- when the real problem could simply be your situation.

Public sector jobs are not inherently less stressful, no matter what people think. Certain aspects are actually more stressful than the private sector.

I'm not sure how big your public sector is, but would a lateral move improve things i.e. a transfer to a different department or a completely different function? It sounds like you have a gift for facilitation and presenting ideas. Assuming that's something you like doing, perhaps you could find a role that would let you do more of that than the thing you think you are not good at?

The thing that you feel you do not do well at your current job -- is it something you *want* to do? Do you see yourself using the skills you would gain by completing the task in a future position?

One benefit of looking for a new job is that it sometimes starts to make yours look better, which can be motivating