Welcome to the forum.
It sounds like you issue is motion blur, and not a focus issue. The shutter speed controls how motion is captured in a photo. The motion may be something moving in front of the camera, or the movement of the camera itself.
So if your photos are blurry, my first question would be; What was the shutter speed?
Even when maximizing your aperture and/or turning up the ISO, you are not guaranteed of getting a shutter speed fast enough to get a sharp shot. It all has to do with how much light you have. Of course, the degree of motion will determine just how fast the shutter speed needs to be.
The rule of thumb for controlling blur from camera shake, is that you want a shutter speed of at least 1/focal length. So if you are shooting at 55mm, you will want a shutter speed of at least 1/55 (1/60). This rule goes back to the days of film, so some people include the 'crop factor' in the equation (1.6 in your case). So you would really want a shutter speed of at least 1/88 (1/90).
When shooting in dim lighting, you may have trouble getting those speeds while maintaining proper exposure.
If you are shooting moving subjects, like kids etc. then you may want something faster like 1/125 etc as a minimum.
One of the main limitations to getter a faster shutter speed, is the maximum aperture of the lens. That is why many of us like to use 'faster' lenses. (larger max aperture). A common suggestion for Canon/Nikon is a 50mm F1.8 lens. At F1.8, you can get a much faster shutter speed than at F4 or F5.6.
Upping the ISO does help, but at the cost of increased digital noise.
Of course, another way to eliminate blur caused by motion, is to eliminate the motion. So if your subject is already still, you just have to steady the camera....a.k.a. use a tripod (or something) for support. When using a tripod, you can use any shutter speed and get reasonably sharp shots of stationary subjects.
Lastly, if you can't use a faster lens, higher ISO or tripod etc., the alternative is to add light. Flash is the most common way. You say you don't like the results from the 'harsh flash'...well then, I'd suggest learning ways to make your flash less harsh. Using flash to your advantage is a very wide topic and there are many techniques you can use to get the results you want.