I always back up the whole folder. If you have the space and don't mind waiting a few minutes then why not? No harm at all.
You only briefly mentioned your old computer. Unpleasant noises can be as simple as fans going out.. which are easy to replace but if it's a laptop then nevermind. But of course a mechanical drive can easily go out too making noises.
My husband's desktop sounded like a lawn mower being started. I bought him all new fans and told him don't bother narrowing it down. It was the fans in this case.
