If you are trying to shoot a sport like football at the typical high school field then you either need a fast (wide aperture) lens and a camera that works well at high ISO or you will need to choose your spot carefully so that you are only trying to capture action in the best lit part of the field AND you can have your zoom set in the range that offers its widest aperture.
Most high school field lighting is very bad but there are some that are decently lit so if your home field is horrible then away games may offer a better opportunity.
The first photo was shot at the school's home field which has lighting standards well back from the field and many of the individual bulbs weren't functioning. I like to keep a shutter speed of 1/1000 minimum for football and shooting with a F2.8 lens still means the required ISO setting is going to be pretty high.
The second shot was from an away game with a very well lit field (from a high school perspective) and with a 2.8 lens shooting at 1/1000 results in a proper exposure with an ISO of 6,400 but even there you would have to drop shutter speed to compensate for your slower speed lens and ISO limitation.
Both of these were shot with 300MM F2.8 mounted on a 1DX II. At the home field I shoot in manual mode but with auto ISO and depending upon the field position, ISO is going to vary from 6,400 up to 51,200 which I have set at the maximum for auto ISO. At a well and fairly evenly lit away field I will shoot in manual and leave the ISO fixed on 6,400 or 12,800 depending upon the field and adjust for variance during RAW processing.
"I am using a Canon EOS Rebel T3 with a Canon Zoom Lens EF 75-300mm 1: 4-5.6 III."
OK, here is the 3 most important things you need to consider. They are location, location and location. Yeah, it is where you shoot from more that what gear you are using. The best affordable sports lens for most of us is the Tamron SP 150-600mm f/5-6.3 Di VC USD G2 for Canon. It isn't any faster, really, than the lens you already have. It is way better however. Unless you have a spare wheelbarrow full of cash you can't buy a better lens for sports.
Next, when I shoot sports, I use Av and I fix the lens at a more open aperture I.E. f5 or f6.3 (for the big Tammy). When conditions become more challenging, you need to abandon the 'auto' modes like 'Sports'. You tell the T3 what you want it to do. With Av you fix the aperture and let the camera select the fastest SS it can. Set the ISO at 1600 but don't be afraid to change it if necessary. That is the limit of the ability of the camera. All cameras and lenses have a limit to what they can do.
Back to the most important part, location. Where are you shooting from? The sidelines, good. Able to roam the sidelines, great. From the bleachers, bad. Lastly, know the game. That's right learn how the game is played. This way you act and don't re-act to the plays. If you constantly re-act to sports, you will never be a good sports photographer no matter how good your gear is.
My personal gear I use for most sports is my 1DX and Sigma 120-300mm f/2.8 DG OS HSM Lens. I don't have that wheelbarrow full of money any longer!