Can I add a utility sink which utilizes the existing water and drain lines for my washer?
You can definitely utilize the existing water and drain lines for this; here is a video from This Old House where they do exactly that:

https://www.youtube.com/watch?v=h-tUGX3M6zY

Your case will probably be easier since it's not in the basement.  Richard in the video re-routes some of the copper plumbing there - I'm not experienced enough to tell whether you can avoid that, but I imagine you should be able to get a splitter like this:



What you do for the drain is use a Y-fitting:



You will also need a couple other pieces; watch the video and it will show them.

Good luck!