I am the appointed executor (through the court) of cousin's will. In the will, she left the house and furnishings to her caregiver. How do I do this

As your probate attorney should advise you, you can record an Executor's Deed (sometimes called a Distribution Deed) in the county deed records transferring title to the caregiver.  Depending on the value of the furnishings, you may want to document their transfer as well.