Indoors with all the lights on at ISO 1600, you should be able to get photos. That's even without a flash and handheld.
As others are asking for with example photo(s) with EXIF intact, the particular info we need at this point is shutter speed and aperture settings for the shots that didn't turn out.
If your images are too dark at ISO 1600 with all the lights on, it sounds off hand like you need to first open up your aperture as wide as it will go, then slow down shutter speed.
No matter what camera and lens you have, the exposure triangle is the exposure triangle, and that's why light meters work independent of type of camera and lens and film or digital ISO used. A set amount of ISO, shutter speed and aperture provides essentially the same amount of shooting light to work with, no matter what. That means if I can shoot it on one of my cameras at X X X settings, you can shoot it on yours with those same settings. Quality and noise and stuff like that may change from sensor to sensor, but the amount of light that comes into the camera and shows on the photos won't, at least not to a significant degree.
Example: This is my laundry room, which is VERY dark. No additional lights on, just a little light peeking around the corner and a little coming in from behind me. My camera is set to ISO1600, my aperture on this 10-22mm is open all the way to f/3.5, and to get enough light, I'm shooting at just 1/4 of a second on the shutter. Because my shutter speed is so slow, I've braced my shoulder up against the door frame to steady myself, and I'm paying attention to keep the camera as still as possible while I shoot. Result: