The Veterans Administration is on its way to becoming the government's 14th full-fledged department now that Congress has signed off on legislation to raise it to Cabinet-level status. The final bill worked out by negotiators from differing House and Senate versions was approved by a Senate voice vote Tuesday after clearing the House in similar fashion. President Reagan has said he will sign it into law. In related action as lawmakers pushed toward adjournment, the Senate adopted and sent to the House Tuesday a bill that would give a cost-of-living increase to veterans and would compensate Vietnam-era soldiers suffering from cancer as a result of exposure to the chemical defoliant Agent Orange.