As a recovering people-pleaser I can understand to an extent where you are coming from. The fact is, no matter how skilled/talented you are, you will not please all the people all of the time.
First, review your job description and match it against your skills/experience. If they match really closely or exactly, then you should feel confident that you have the capability, no problem. What seem to be happening is you doubt yourself and the anxiety field you create around you probably makes coworkers uncomfortable. If you can do the job, you can do the job, you don't have to prove anything.
I have several suggestions to help you. First, continue to use positive self talk and reassure yourself as often as possible that you are good, worthy, skilled and competent. Do this especially during times of stress/anxiety.
Also, Jack Canfield has a great audio program on this. (
How to Build High Self-Esteem: A Practical Process for Your Personal Growth Audio, Cassette – Audiobook, Unabridged) He is an expert in self-esteem improvement and the tapes are very inexpensive online (~$4) or free at your local library.
Finally, I recommend a spiritual practice to help you. I'm not trying to preach at all, but my personal testimony is that God has graced me with a lot of help to overcome my personal and personality problems that have caused me much difficulty. I've had a lot of help from spiritual mentors to keep me on the path and accountable.
If you want to know more or if there is anything I can do, please let me know.