I fear taking responsibilities. I always feel that I will mess up with the given work. This feeling is very intense and I always tend to find excuses for not doing the work and also it becomes very difficult for me to socialize with new people in new surrounding due to this I avoid any hangouts. Is this anxiety ?
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The problems you have decribed here seem to be pointing towards what we call "Avoidant Personality Disorder" or an Anxious Avoidant personality. The classical features of which are a predilection for tension, self-consciousness, fear of negative evaluation by others, and emotional insecurities.
The other possibility is you have anxiety issues. If your anxieties are restricted only in social situations like public speaking, or meeting new people, or going to new places, it could be Social anxiety disorder.
Irrespective of the couple of possibilities I have mentioned above, what you need to do, is seek a detailed psychiatric evaluation from a Psychiatrist of your choice. The reason for that is, many things in the clinical history of a person, which are vital to arrive to a correct diagnosis, can happen in direct, face to face interview. A medium like this, poses limitation on my understanding of your problem and the advice I can give to you.
So, take a consultation at the earliest and get the issue sorted out. I'm sure it can be easily taken care of.