In this case no...the rent was paid. The landlord had no right to consider the unit abandoned when the rent was paid.
That is absolutely incorrect.
Abandonment has very little to do with whether the rent is paid or not. It has to do with whether the property is occupied and as such, less of s concern it could be broken into or there could be running water in the premises that could cause damage. Additonally it is concerned with when a landlord can reclaim a unit to prepare it for rental to a new tenant.
In the case at hand op said they would leave BY the end of the month. They left. Apparent attempts to contact them to inquire whether they had vacated went unanswered. Is a landlord supposed to just guess the tenant is coming back for anything?
Btw; Utah requires a 30 day written notice of termination. That means op would have owed as many days into October as there were days in to September when he gave notice. It doesn’t sound like op was intending on paying for that time. (Just my read on the statements provided)