That behavior is not normal. In Lightroom, you should click the eyedropper once to pick it up; the tool icon area becomes empty and your cursor changes to the dropper. You can then move over the image and click where needed, such as a neutral gray area for white balance. When you’re done, return to the tool’s circle and click again to put it back.

If it’s switching off by itself, first make sure it isn’t a mouse or pointing-device issue. If your hardware seems fine, try resetting Lightroom’s preferences/settings. If that doesn’t help, reinstall Lightroom. If the problem still persists, it may be a bug and you should contact Adobe support.