I think you're over-complicating things. When you find the differences between the responses before and after the treatment, the group of the individual (whether case or control) doesn't change. So, you can apply t-test on the diff variable using the existing Group variable.
I'll suggest you to check the function by, if you're planning to test manually. Otherwise,t.test is great.
It looks like a assignment problem, so I'll suggest you to get familiar with the homework policy: