1. I'm sorry to bring up my resignation at this moment.
我很抱歉在这个时候提出辞呈。
还能这样说：
I'm sorry to submit my resignation at this moment.
I feel bad to resign at this moment.
应用解析：
the moment (that...) (=as soon as) 一……就……；
moment of truth （接受考验）关键时刻

2. I feel very sorry, could you let me resign?
很抱歉，您能不能批准我辞职？
还能这样说：
I'm very sorry, I'd like to resign.
I feel very sorry. Could you give me permission to resign?
应用解析：
resign oneself to extinction 束手待毙；
not resign oneself to 不甘心

3. I've made a tough decision. Here is my resignation.
我做了一个很困难的决定。这是我的辞呈。
还能这样说：
It's a tough decision to quit this job. It's my resignation.
I've made a tough decision to quit this job. Here is my resignation.
应用解析：
act with decision 断然行动；
hand down a decision 作出判断

4. I want to resign my position as manager.
我想辞去经理的职位。
还能这样说：
I want to quit my position as manager.
I'd like to resign my position as manager.
应用解析：
take up the position 主张