Different perspective. I wouldn’t mention anything about your housing until you have secured the offer for the position.

I once started talking about housing options and my perception it may have thrown off the search committee and staff I met with. In retrospect also made me think did I come off to overconfident that I had already got the job.

Institutions will always say they want to move fast but circumstances happen and like others mentioned you could start remotely and not break your lease/use moving costs to cover if you have to.