How do I stretch a school budget across every need?

You can manage the budget, and still it will not cover everything, because it was never built to. That is not a failure of your skill. It is the honest math of too many real needs and too little money. Feeling defeated by that means you are seeing clearly, not falling short. When you cannot fund everything, the job stops being math and becomes values. You are not looking for the arrangement that meets every need, because there isn't one. You are deciding, on purpose, which needs come first and being able to explain why. A budget is a moral document. It says out loud what you have chosen to protect. Before the next round, write your top three non negotiables, the things that get funded before anything else no matter what. Let the rest compete for what remains. This will not make the shortfall disappear, but it turns an impossible task into an honest one, and it lets you sleep, because you chose with your eyes open instead of hoping the numbers would save you.