and tried to dispose my duties- it should be present perfect- have tried- but why not be a bit more confident- and have performed my duties to the best of my ability.
I hope my sincere efforts have been justified by my obedient and punctual working.- This sentence doesn't really fit to me- if they have increased your responsibilities, then you have done this
to be more responsible in my assignments- this suggests you have been less responsible in the past, so try this:
I accept the greater responsibilities and shall handle them with the same dedication and attention.
Thanking you in advance, I look forward to hearing from you.