Dear Vany:”

If “they” are your co workers, not your supervisors, instead of telling them (your quote):
“I’m sorry. But my plate is really full today, I’m unable to do the favor you ask me to.”

I wouldn’t say I am sorry at all. I wouldn’t apologize, that is an unnecessary gesture of submission in this context. I would say instead: “No, I will not do that.”

Back to your quote:”Their response; ‘No, don’t worry, you just call them after this thing you’re writing.’
Me; ‘No, I told you I don’t have time, my client is in a rush and I’ve got a deadline.’”

Instead I would say: “No, I will not do that” I will not tell the co worker that you don’t have time etc. that suggests to her that you think you are supposed to follow her orders, only that you don’t have the time right then to follow her orders. The point you want to make to the co worker is that you are not submissive to her and you don’t follow her instructions, or orders.

anita