A 35mm prime lens on a D3200 with an SB-600 should be able to take pictures in the dark. It sounds like the tools you already have aren't being used to their full potential. It should go without saying that you should check to see that the batteries in your flash unit are adequately charged.

Without an example or two, and based only on your description, I'm guessing that you were using a flash setting that included slow sync to take the photos at the baby shower. There are two problems that must be overcome to successfully use slow sync: camera motion and subject motion.

In a dim environment when set to slow sync the camera will slow the shutter speed enough to properly expose the background using the available ambient light and then add only enough flash power to properly illuminate the subject in the foreground. To prevent blur from camera shake in such a situation the camera needs to be mounted on a solid object such as a tripod. The subjects also need to remain still during the entire time the shutter is open. This isn't very practical for a social event such as a baby shower when you want to capture candid moments.

To get better results in a dim environment, set your D3200 to A shooting mode and select a fairly wide aperture such as f/1.8. Be sure your flash is set to a mode that allows i-TTL to be active and that it is not set to a mode that includes slow sync (see pages 48-50 of your D3200 Reference Manual for details). Check that the exposure compensation and flash exposure compensation are set to '0'. Select an ISO that allows at least a 1/60 sec. shutter speed and work from there. If you are still having problems, change to S mode and select 1/60 sec as the shutter speed. If the camera tells you the largest available aperture is not wide enough, increase the ISO setting until it is.