It sounds like its a matter of positioning focal points on exactly what you want to shoot when you're shooting. It also sounds like your f-stop is wide open, which makes your focal length very short. Try setting it to 5.6 or so. Your images will still have the beautiful crisp sharpness of the 50mm prime lens, and MORE depths will be IN focus that way.
To give you an example, lets say you're standing 5 feet away from someone with your F-stop set to 1.8. If you very carefully focus on their eyes, it is very possible that their ears may be slightly out of focus because the focal length is THAT short. Its fine and good to do that, just be aware that you have to be VERY precise with this method, and it is of course best to always focus on the eyes. Just a note as well, that the further back you are from your subject, the more apt you are to have more area in that tack-sharp focal area.
(Another note that the blur you're seeing is referred to as "bokeh" - the ever-desired blur for the wide-open f-stop that you get with prime lenses. Its beautiful in background to make your subject pop out of the frame, but when misplaced can appear "blurry" rather than done on purpose)
HOWEVER, if you set your f-stop to 5.6 or above, you will likely have their entire head in focus, and be more likely to have that tack-sharp look over more depth. To better explain this, you could have sharpness from the front of the nose all the way back to the ears without a problem, and your background will still have some bokeh (though it won't be AS much).
A good way to play with this is to set your camera settings to "aperture value" (AV on Canon cameras - on Nikons it is simply A). Then you can manually adjust your aperture, testing out the different focal lengths, while your camera adjusts appropriately for shutter speed and ISO.
ETA: It might be a good test for you to set up a scene (say a bowl of fruit on a table or something else that won't move), and try taking pictures of it one after the other with different aperture settings to see what the difference is with the blur (bokeh) and focus effects. Just make sure to have good lighting (such as a wide north-facing window or in an open garage) so that your shutter speed is at least 1/100 of a second, since anything lower can take away from that "tack sharp" look you're going for. You can help this by upping your ISO if you see your shutter speed is going below 1/100 on an automatic setting.
Hope that helps a bit.