Can I be refused renting a home due to my first DUI charge?

Question Details: I got a DUI 5 months ago and we just went to court; it's now showing up on my credit and criminal record. I am in the process of trying to move to a new rental home in another town nearby. Can the DUI charge give the landlord the right to not rent to me even though I have paid my rent on time for over 30 years and never had a DUI before? If they stop me from renting then what do I do? Can I get around it? My lease is up at the end of this month and the place is already re-rented.

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