You are describing the stems that the handles attach to.
I would try to extract the screws from the fine threads inside of the stems. A small drill bit followed with an extractor should remove them. Retrofit handles can be used, but I am not too fond of them. They seem to not grip to the stems too well since they are universal to fit many sizes.
To remove the stems you would have to shut the water off to the unit in order to replace them. If you are renting, I advise letting this job go to the maintenance that is the responsibility of the landlord.
DUNBAR
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