The Sony 90mm Macro 2.8 is one of the older lenses in the Sony setup. It is superbly sharp, but the tech has aged a bit and the AF is not the fastest.

The stuff that you describe can happen if you are in AF-C on a smaller aperture and/or it is too dark - or you have the limiter on and the AF does not directly establish focus.

The focus limiter is added to the lens specifically to limit the area through which the lens will hunt for focus. So you can choose to either include the full range or limit the search range to only far distances beyond 0.5m or to very close to speed things up.

But in case you misconfigured this, even hunting will not lead to finding focus as you excluded the needed distance from the allowed focus range. So I assume this is not the problem here.

This "focus-hunting" could also happen on subjects that simply lacks contrast to find focus on.

Please check if you also have the behaviour in good lighting condition on AF-S.

On AF-C you stay on the set aperture, which means less light for the auto focus, while on AF-S the lens will go to f2.8 for focussing and then switch to the desired aperture after finding focus to help the AF.

If this happens in good light with a clear contrasty image, then I would consider letting a shop have a look at the issue.

In external reviews like this one https://www.talkemount.com/threads/sony-fe-90mm-f-2-8-macro-g-oss-lens-review.13173/ the reviewer also states, that he enjoys the lens, but the drawback its tendency to hunt for focus in lower light. This also was my own experience.