Would a broken sway bar cause catastrophic failure like when a tie rod breaks? I'm not very familiar with suspension components other than shocks and springs.
A broke sawy bar will not cause anything catastrpohic. The car will not stay planted to the road as nicely as it should when making agressive turn's \ corner's. Nothing to worry about at all. None of my Drag Car's run Sway bar's and I drive them on the street more then enough with zero issues. But on a Daily Driver I'd look into getting the item replaced if that really is an ungoing issue. But I doubt thats what the issue is with the sympthoes you are explaing. Did you buy the car new and it had this issue or used?