How to fix a sloppy safety?

[DonVoigt]

Ollie,
Is it a "shooter" or safe queen?

Safe queen- I'd take the safety lever out and through a couple burs on the rotating section or the safety- the "axle", or shim it at the bottom of the hole as you mention.

On a shooter, bend the exposed part of the lever-the thin part- by holding at the axle and bending the thin, springy part only. For a more permanent fix, heating-quench-temper would be better, but you don't know how hot to heat, what to quench in, or what temp to draw to.

Then it could be the little "nub" is completely worn away that stops in each of the recesses, then another fix is in order!