The Tamron is known to be optically very good and sharp wide open across the frame. I know semi pro Nikon users who use that one on a crop camera for e.g. wedding shots. The F number on your kit number is only F3.5 on the widest and if you go into Av and keep an eye on it, you see it drops very quickly to F5-5.6. The range 2.8 - 4 is a stop (double the shutter time - F3.5 vs 2.8 is a factor 1.56) and 4-5.6 is another stop (quadruple the shutter time).

Now the thing is that the longer the focal length the shorter the shutter needs to be so it is working against itself. at 18 you can do fine with 1/27s, maybe 1/40s to be sure, while at 50mm you need 1/75s minimum or 1/125 to be sure.

Another problem is that if you get serious and try to control the composition including depth of field, ambient/flash light ratio, exposure manually, you have to worry about zooming just a bit as it might change your balance.

Also if you have a filter on that makes everything darker, a Fx-5.6 lens will go darker than F5.6 for most part of your zoom range, disabling the AF which doesnt work under F5.6.

So this is all the trouble you are relieved of in a fixed aperture lens.

If your most important objective is sharp indoor photos without flash, you might need to sacrifice the convenience of zooming, and go straight for F1.4 or F1.8 primes. That's 1-2 stops faster yet from the F2.8. A F1.4 50mm is 16 times faster than your kit lens. When I shoot indoor in normal living room light I already have to use Iso 800 or 1600 depending on the light and whether I use 28mm, 50mm or 85mm. I do stop it down a bit to F2.2 to improve sharpness a bit and get a bit more focus in DOF and that gives a fast enough shutter (1/50s - 1/150s).