Well, if you can get reasonably close to the field, at ISO 1600 you could probably manage a shutter speed of 1/350 to 1/500 second at f/1.8 with a fast, short telephoto lens like the Canon 85mm f/1.8 EF lens. It is a lightweight, SHARP, fast-focusing moderately long telephoto lens. Not sure what position he plays; receiver, running back often "come open" pretty close to the sidelines and end-zone areas so if you are say, right down by where the track meets the corner of the endzone, with an 85mm lens, you can get shots of TD passes and TD runs clear over to the OTHER side of the field. At one time, an f/1.8 lens was considered extraordinarily fast; even today, it is the second-fastest telephoto lens most people will ever own.
The real problem is the limited ISO of 1600 AND the slow f/5.6 limit of the lens you have. The 85mm f/1.8 is my recommendation for all beginner and intermediates who want a lens for LOW-light, distance work. It is the ***best*** overall value you can get in terms of what it brings to the table for the money spent. Your camera cannot be cropped a lot, so the 85mm length makes it worth the $400 or so to buy an 85/1.8. It is also NOT too long, so you can "find the action", and at longer distances of say 50 to 150 feet it is SHARP, and the focusing is good, and fast. It's not a 300/2.8 or 400/2.8, but you're not a pro sports shooter. A lot of hardcore, experienced shooters suggest longer, $7,999 lenses, but they forget how impossible that kind of lens is to use for a beginner or intermediate shooter used to a 20-ounce lens.