Hello, Itdeleon!
Unfortunately, it looks like you are approaching the limits of your gear. IIRC, the T3 ISO maxes at 6400 and you are halfway there with the blurry example you provided and noise generally increases from there. It is blurry because your shutter is too slow for the action (see attached). You may be able to raise your ISO or even max it out, but that may or may not be acceptable or even get your shutter speed to where it needs to be. All you can do is try and see if the shots are acceptable to you.
Newton
A tripod isn't going to help with this type of blur because the shutter speed needed to freeze soccer player motion (1/1000 or better preferred, 1/800 fairly usable, 1/640 will result in a fair amount of blur) is far faster than needed to avoid blur caused by camera shake.
Typical high school soccer fields are not brightly illuminated so, like HS football, it is a sport that requires a wide aperture lens. I believe your current lens wide open is f5.6 at the telephoto end and even with a really good high end camera body, a f5.6 lens would make things difficult to get fast enough shutter speed while keeping the ISO reasonable. A body that handles low light well and a wide aperture lens are both critical to night sports in high school but the lens is the most critical and if you decide to upgrade, that is the direction to go first.
It will be limiting but at night your best bet is to get as close to the action as possible on the best illuminated portion of the field and keep your lens within its focal length range where f4 aperture is available. Try experimenting with 1/800, 1/640, and maybe as low as 1/500 (although this is going to introduce some major motion blur from player speed) to see what works best for you. Unfortunately high school sports is actually more demanding of equipment than D1 college or pro-level because at those venues the level of illumination is much higher.
Example photos below were shot on an average HS field using f2.8 lenses (70-200 f2.8 and 400 f2.8) with a pair of 1DX III bodies at 1/1,000 shutter speed with the lens wide open. This resulted in ISO 10,000 for all except the second photo which was in a relatively bright spot at ISO 5,000. In daylight or on a well illuminated pitch, a f5.6 lens will do fine but not so much in the dark. The fifth photo was at an afternoon match and I shot this game using my EF 200-400 f4 with integrated f1.4 extender; the last photo was captured while I was at the opposite goal with the extender switched in resulting in a 533mm f5.6 lens but due to plenty of daylight ISO wasn't an issue.
Rodger
