Who are you shooting for?
IMHO, you will struggle to meet your budget.
Your primary issue and constraint is LOW light.
There is a high school where I will NOT shoot a night game. The field is too dark for my gear.
That is what I mean by LOW light will be your primary constraint.
Camera:
Find the camera with the best LOW light performance. You want the max ISO AT LEAST 25600.
The easy method is to see what the max ISO is. If you need ISO 25600 or 51200, and you only have 6400 or 12800, you are stuck.
The Nikon Z6ii would be my choice.
I am not a Canon or Sony guy, so no idea of their specs. You have to research them.
But . . . Canon has slammed the door on 3rd party lenses for their RF mirrorless cameras. Which means only expensive Canon RF lenses on the RF cameras. The Canon RF cameras will kill your budget on the lens.
dSLR alternative are used dSLRs with a high max ISO, of at least 25600.
Lens:
"In LOW light, FAST glass wins."
For LOW light, you REALLY NEED a f/2.8 lens. Even with a camera with a high max ISO.
Which lens depends on WHERE you are. IOW how FAR are you from the field.
On the sidelines, a 70-200/2.8. For your budget, I suggest either a Tamron, Sigma or used Niikon/Canon/Sony. Check
KEH for prices.
Up on the bleachers, a 300/2.8 or 400/2.8. But, the 400/2.8 is EXPENSIVE. A used 400 is over $5,000. So forget the 400. You will just have to use the 70-200/2.8. Or try to find a used 300/2.8 or Sigma 120-300/2.8 at a price that meets your budget.
My recommendation would be a used (
KEH) Nikon Z6ii + Nikon 70-200/2.8. That will just about fit under $3,500.
But that leaves you with little/no extra $ for a short lens, for shooting the sidelines.