Typically, unless cutting politics updates, I'll cut evidence with a certain argument/file in mind. I'll start off with engines my school has to offer. We had lexis and proquest and a few others. Then, I'll go to Google, but start on Scholar. This will get lead you to a bunch of articles in books and other Scholastic articles. If all else fails I'll usually end up going to the Google Search Engine and spend hours upon hours looking at a lot of junk.
I don't know what your operating system is, but I use Windows, so I'll be talking about things I do in there. To speed up cutting, I'll have a notepad window open so I can unformat text from the internet. You should also make macros in Word. The click of two keys on the keyboard is certainly much faster than bolding, changing size, or underlining with the mouse. Do it all in two strokes. Also, when you find scholarly articles that link to or have other articles in the bibliography, you will more effectively find evidence in those places.