The two-way fixed effects regression is the default difference-in-differences estimator in applied economics — until treatment timing varies across units. Once some units are treated earlier than others, TWFE implicitly uses already-treated units as controls for later-treated ones, and those comparisons can get negative weight.
That means the sign of your reported effect can be the opposite of every single unit-level effect in your data. Not a small-sample fluke — a structural property of the estimator under staggered adoption.
The fix isn't a different standard error. It's a different estimator: Callaway & Sant'Anna's group-time average treatment effects, the Sun & Abraham interaction-weighted estimator, or Borusyak-Jaravel-Spiess's imputation approach all avoid using already-treated units as comparisons. Pick one, and check whether it changes your headline number before you trust the TWFE coefficient at all.