What is the proper air strip closure procedure with the FAA?
It looks to me like you need to file FAA Form 7480-1, Notice of Landing Area Proposal.  There is a checkbox for "Deactivation or Abandonment".

The form explains that you have to give 90 days advance notice.  So you shouldn't make any changes until 90 days after filing the form.  This presumably gives them time to update charts, so that pilots know the airstrip is no longer usable.

The form has the address and phone number for your FAA regional office, who might be able to help if you have questions about how to complete the form.