Odds of Getting In
There's good news...and there's bad news. While you have what we like to call "an actual skill" (sorry, Miranda, that emo-goth beat poetry you wrote freshman year cannot be used on your résumé), there isn't the same demand for it as, say, app development or engineering. So while you may be a great candidate for the jobs that are out there, you'll likely have to do some sniffing around to find them.
Just showing up at a police station with a picture you drew of the chief's mom might sound like a good idea, but it probably isn't. For lots of reasons. Many times, stations hire internally, or only use artists who have a proven track record of results in a similar field. That means either getting creative and assertive, or sitting around online all day calmly hitting the refresh button on your job board of choice.
If you're serious about sketching professionally, don't lose hope. There is work out there—just don't expect to fill out a one-page job application, meet with a manager, and start forty hours a week on Monday.