That way if there is another offer you won't get caught up in the emotions should there be a bidding war. 

The logical answer is terminate the lease and the emotional answer is to let them stay.  

I recently rehabbed a property, signed a 2 yr lease with someone, then about a week before she’s supposed to move in, she informs me that she has an ESA (Emotional Support Animal), a huge dog - lab mix.

Got emotional when they said there were other offers higher than mine.

You really have to not allow emotion be part of the equation when dealing with certain types of tenants.

I recognize emotion and fear when I'm in the process.

Don't let emotions get involved.

That may be true, but having an emotional attachment to an investment doesn't mean you are best suited for managing it. 

Is anyone else seeing a weakening in the rental market? 

With knowledge FEAR becomes nothing more than a irrational emotion.