Going up against the Federal budget

There has been a lot of talk about hiking the federal debt limit, something that will become pressing this spring when federal borrowing bumps up against the limit. If the Republican house wants to use this crisis constructively, they could say "After we pass this year's appropriation bills, so we know how much deeper in the hole we are going to go, THEN we can talk about OKing more borrowing."
The Democrats adjourned last year's Congress without passing ANY appropriation bills. The whole federal government is running of a "continuing resolution", which runs out about the same time as the debt limit will be reached.
In short, use the debt limit as a lever to lower overall federal spending.