I have never once heard during the budget negotiation process, "we cannot increase this budget because it will put us over the debt limit."
The debt limit restricts the US from paying for things it has already purchased. It in no way stops the US from purchasing them, or budgeting for them, in the first place.
Using the debt limit as a means of controlling future budget negotiations (in other words, "adopt my budget priorities or I will tank the economy") is exactly why Fitch downgraded US debt, thereby increasing the cost of debt service and exacerbating the problem you're concerned about.