Is Papaya King finally out of the doghouse? The renowned hot dog joint on New York’s Upper East Side shuttered last year and was trying to reopen at a new location across the street until late last year, when the arrangement landed them in legal hot water. But now owner Sajid “Sid” Sohail and the landlord of the Third Ave. property where the eatery tried to move into have dropped the lawsuits they filed against each other over a lease disagreement, according to court records. Papaya King closed last spring after 90 years on the corner of 86th St. amid legal troubles and the threat of eviction,…