It sounds like you are underexposing. Noise is generally a more significant problem when the brightest pixels in your image are only about 18% gray level. If this is the case on a consistent basis, especially if you are encountering blurring (I assume due to camera shake) then you are plain and simply using too low of an ISO setting.

If you use a low ISO setting and under-expose, then boost that exposure in post, your actually amplifying the effects of noise a lot more than if you use a higher ISO setting. If you are getting blurry, noisy shots at ISO 400, try ISO 800 or even ISO 1600. The goal is to saturate your exposures, pushing the average per-pixel luminance beyond that 18% gray level. The more pixels you can saturate beyond an 18% gray level, the less noisy your image will appear. If you over-expose but do not blow out your highlights, you can recover exposure in post with a little negative exposure compensation or highlight recovery.

The fear of using a "high" ISO setting is often the cause of noisy photos. High ISO is not bad, and in low-light situations (even with flash), you need to use the highest ISO setting you must as dictated by minimum shutter speed requirements. In general, the camera will do a better job of boosting exposure while minimizing noise when using higher ISO settings than you could generally do by shooting at a lower ISO setting and trying to compensate for underexposure in post.