Both are problems! The FED is easier to solve. Shutter it, return to coinage of standard weight, and notes payable in same, as the Founder’s established in the Constitution.
Only a rare and temporary Congress will have the fiscal discipline pure fiat money requires. Reasonable men do not rely on such rarity for the regular operation of government. To do so is insane, immature, delusional.
Congress can borrow and spend beyond their means on a gold standard just like with fiat money. And a fiscally irresponsible congress can and will promise that your children will repay their debts in gold.
I've already shown a graph that Congress reached even higher levels of debt to GNP while on the gold standard than current levels of debt.