The word responsibility is one I strongly identify with--to the point of placing my significant value in how responsible I was that day/week. Of course, responsibility is up for various interpretations and those that toot their horn and say, "I'm so responsible," often take on the tasks and obligations of others--even when they shouldn't.
Although I cannot speak for everyone, at the core of my issue is a need to be in control. I lack the trust in others or God, thinking that it's all up to me. "I've got to be responsible. Someone has to do it, and you know it's going to be me. If I don't do it, no one will." Or "If I don't do it, it won't get done 'right'." Ultimately, I am playing God and taking myself far too seriously. As someone so appropriately worded, "When I take myself too seriously, I am not taking God seriously enough."
When I find that I have become too "diligent," the practicality of life suffocates any sense of humor or playfulness. I forget how to just have fun (and not feel guilty); I forget how to laugh at myself (because of course there is no room for error), and overall, I am a miserable person (to others as well).
So, in an attempt to regain my sense of balance, I have to shift my perspective. And recite to myself, "I am not that important. God is still God; I am not. And it's okay if it isn't done perfectly or if I take a little break."
For those that really struggle with delegation, because they are afraid of how the task may be completed, Norman Vincent Peale suggests visiting a cemetery every week. Many of the men and women resting there once believed that the world rested on their shoulders, but life continues on. And the only one who continues to remain is the Lord God.
Here is another suggestion for you when you find that you have become too serious and have lost your playful ability to relax. Pick up a coloring book and a box of crayons (yes, by yourself) and color a picture for ten minutes. It's incredible how the pulse will slow and the tension will ease when we do something childlike. Don 't think about all the tasks you have to do or the guilt that is nagging at your subconscious; just let yourself think about the picture, and the colors, the wonder that God gave us color and the ability to imitate creation. We are imaginative creators who can play and laugh and enjoy life--gifts from Him (not the enemy). So let yourself play, and remember not too take yourself too seriously.