That is exactly how it must be done! An omnibus spending bill allows the president to "shift" the uses of that whole pot of funds. But if Congress passes individual bills allocating limited funds for specific departments and agencies, the president can only spend those amounts toward the uses for which they were appropriated.
The House of Representatives holds all the cards, since it alone may originate spending bills. What remains to be seen is whether or not the House has the guts to play its hand. Whig status awaits if the GOP folds...
That what was wrong with the government shutdown in 1995, they tried negotiating an omnibus budget with Clinton and threatened to shut down the government. This gave Clinton a huge political opening, through which he ran. This time pass the appropriation bills separately and frugally and let the dems stop, veto or whine.