The Tamron 17-50mm f/2.8 XR Di II is probably a better fit for your Rebel T3i than the Tamron 28-75mm f/2.8 XR Di. With the 1.6x crop factor the equivalent field of view (FoV) is for a 27-80mm and 45-120mm lens respectively on a full frame body. The 17-50 is also a newer design that is optimized for cropped sensors. I own one and it is quite a sharp lens for the money.
If you compare ISO 12233 tests at The-Digital-Picture.com, the Tamron 17-50 is sharper than the EF-S 15-85 at the focal lengths they share in common, even when the Tamron is opened up wider than the maximum aperture of the 15-85. Both these lenses are sharper than the Tamron 28-75, which has become a rather dated design that was good enough for film but not quite good enough for pixel peeping in the digital age.
Some people complain that the Tamron's focus motor is noisy and slow. It does make a unique sound while focusing, but I think the sound it makes fools a lot of folks into thinking it is slower than it really is. It focuses about as fast as any Canon lens in the same class that I have used.
The Tamron 17-50mm f/2.8 XR Di II runs about $500, the Canon EF-S 15-85mm f/3.5-5.6 runs about $800. So you are paying about 60% more for about 70% more reach while giving up between 2/3 to 2 stops of aperture and a little sharpness.
With your children indoors, I think the shorter focal lengths between 17-50mm is where you will spend most of your time shooting and the constant f/2.8 aperture of the Tamron will be invaluable in the lower light indoors, allowing you to use lower ISO than the f/3.5-5.6 lens will require. Outdoors the extra reach of the EF-S 15-85mm will come in handy. For shooting static objects in low light, the Image Stabilization (IS) will be useful. A Vibration Control (VC - The Tamron term for IS) version, the Tamron 17-50mm f/2.8 XR Di II VC is also available for about $650. The partial review for the Tamron 17-50mm f/2.8 Di II VC at The-Digital-Picture is disappointing, but at DxO Mark the VC version tested slightly better than the original, non-VC version. The EF-S 15-85 tested a little less sharp at the corners than either Tamron at 35-50mm.