Yes, no agreement would effectively leave the US government unable to pay its bills – such as social security payments, Medicare, military salaries.
However, fiscal inflows should cover treasury coupon payments with room to spare for a few weeks and the US government can selectively close itself down in order to preserve cash. Maturing debt can be rolled over. Therefore, America is not about to hit the wall
The biggest potential scenario is that if the crisis continues, and US govt debt is called into default, then this means that banks would technically be barred from using US debt as collateral with central banks or the settlements clearing centres, (which will be Armageddon) although these rules could be changed…. like the ECB is contemplating for Greece.
If you’re that worried, switch everything into gold or Swiss francs.