Again I dont think, this is so much a problem with the camera.
You are using a one-size fits all 'walk-about' super-zoom, lens that is a jack-of-all-trades-master-of-none, 'all-rounder'; it's not a particularly 'fast' lens for low light... which I am a BIT sanguine about.
f16-Sunny, suggests that the 'normal' sort of exposure settings on a 'good' sunny day, are a shutter-speed 1 over the ISO at f16... open up a stop if dim or a bit cloudy, two if very cloudy or starting to rain.
Its actually been pretty bright of late, and with clear skies and low winter sun, on a sunday-morning, it actually shouldn't be 'far' off an F16-Sunny situation...
This begs suggestion that you are using a LOT of that zoom, to get in tight on the action, and pushing the shutter-speeds up to try freeze it... and STILL suffering motion blurr? I am going to hazard that with that much zoom, its as likely to be camera shake....
Rather than blaming the gear-here.. I would suggest taking a couple of steps back, and look at the craft.....
And TOP of the list, SIMPLY not trying to zoom in so tight.
You zoom in tight, you are peering into the shaddows of whats going on; you probably, if my murky recolections of Sunday-morning scrums is anything to go by, have rather 'dark' muddy field as back-ground, and bunch of kids all low down, blocking the light. The Through-The-Lens light meter in your camera, is then likely not seeing as much light as you would looking at the whole scene, and even if you are making manual settings, you are still making them to the TTL suggested light reading... and that's presuming an 'average' brightness reflected off the scene, that you dont have, but on the presumption you do, the vameras electrickery is pulling settings 'down' as if it was.
But, top of the list... USE LESS ZOOM...... you don't need to get in that tight on the action, and it probably ISN'T actually making a better photo. YES it cuts out extraneous detail, concentrates the viewers attention on that little bit of the scene you were interested in at the time, and delivers 'instant impact'... BUT cutting so much 'clutter' from the shot, you also crop 'context'... you want pictures of just your sons face, in the mud holding a rugby ball... why not take him down the park, get him to pose like that, and take as much time as you want and get the exact angle you want, rather than leave it to chance in the heat of a game, and make life hard for yourself trying to set the camera up in a hurry to catch it..... step back.. zoom out... get the conmtext, show the 'game' going on, show the pitch, let viewer see that its not a single kid in the park posed for the shoty, let them see it IS a game, and possibly where it is, and rather than give them instant 'impact' that's as instantly ignored after they have looked.. give them 'interest', MAKE them look at the detail and 'see' the other kids, the action around the subject, the pitch, the whole scene... give them the WHOLE picture, not just a fraction of it.
Zoom out... almost garanteed that at shorter, wider focal length settings, your lens will have lower f-numbers available, and in effect be a faster lens. Small asside here, the F-No is a fraction of the size of 'hole' that is the aperture, in relation to the length of the lens. So, if you have a hole, say 20mm in diameter, behind a 200mm lens, the ratio is 20mm/200mm or f10. Same hole, 20mm in diameter, behind a 100mm lens, now is 20/100, its f5... and if you rack out to 50mm, its 20/50, f2.5.. this is why the f-number changes as you 'zoom'.. there's some complexity created with more complex lens construction where the f-number is actually an 'equivilence' because of how the aperture is placed behind the elements and they all move as the lens is zoomed, but that's the basic principle; and why zoom-lenses are quoted with two 'maximum' apertures, one at the wide end, one at the long... actual max aperture probably doesn't actually change size, but the ratio does as the lens zooms, and its cheaper and easier to give you, the customer that 'faster' aperture at the wide-side, than either try and make the lens keep a constant F-no through the zoom range, by cleaver and probably complex arrangement of elements and iris, or simply 'choke' the iris at shorter zoom lengths so the number stays the same.
But, point is, instant you zoom out... first you will almost certainly have faster apertures available. Second, with the cameras Through-The-Lens exposure meter seeing more of the scene, it stands a better chance of not 'so' easily or so much, being fooled into upping the exposure settings by the dim-bit you show it.
Next.... apart from perhaps getting one or two-stops extra apperture from the lens, so you can more readily run faster shutter speeds; with less zoom, you PROBABLY don't need to run such a high shutter. The subject will not be moving so far in the frame, wither from thier own speed, or from possible camera shake. Old rule of thumb was to keep the shutter above 1/focal length of lens, and for a zoom lens that was oft advised to be the max focal length, so in your case around 1/250th of a second...
Lets look at this.... f16 sunny... lets be generous and add three stops for it being a bit cloudy; Tameron specs suggest that the max aperture at the 270 setting is f6.3.. Canon specs suggest ISO up to 800 in auto,m 1600 in manual, so even there you have a good two-stops more aperture than f16, and you should be able to run shutter-speeds up[ as high as 1/500th comfortably enough....
Even more, this does NOT point to your disapointments being so much of a shortcoming in the kit in your mit, but the know-how in your noddle.... and finding a camera with higher ISO capabilities and or better low light performance REALLY isn't going to solve the fundemental problem here...
So, work on hand holding technique; use the optical view finder, not the back-screen, and hold the camera steady; work on paning to keep the subject inb frame, and be smooth about it... and use less zoom, to giove yourself the 'tolerance' room around your subject as you do it. I suspect that focus, focus lag and focus lock is possibly getting in on the act here, and personally, having grown up with manual focus cameras, switch that ruuddy system 'off' to dodge the issue. snapp down the aperture to increase DoF, and at rugby-pitch sort of range, should be able to cover pretty much the whole pitch with the Depth-of-Field and NOT even have to worry about focus! But... again, zoom out.. you stand better chance of keeping red dot on your target, and giving the lens less to do in such a hurry... and you have the head-room on your ISO you should even be able to stop down a couple of notches and STILL keep the shutter over the focal length, and motion blurr in check.
By all means... if you want a new camera, go buy one... BUT... working with a one-size-fits-all lens, here.... 'must stick to cannon cos lenses'? Eh? you only HAVE/USE one!!!!! That seems a very spuriouse reason for that particular buying constraint! Again, end of the day, your money, spend it how you wish... BUT!!! you go get another camera to stick that lens on... MOST of the problems here will remain, because the problem ISN'T with the camera! Another 'faster' lens would actually make more sense, but still.. the problems noted are significantly NOT limited by either camera or lens... and you would likely get a LOT more, simply learning a bit better craft.... and if you get that, you probably dont need spend a penny... and if you do, with the craft, you might actually get your money's worth from being able to exploit the 'extra' it offers over what you have now, which you probably aren't getting all you could from.
ZOOM OUT... start there, and see where it takes you.