He is charged with simple possession of marijuana and drug paraphernalia and possession of a loaded firearm.
Goodson was the passenger in a car that was stopped in New Jersey after a tow truck driver called 911 to report that the driver, Garant Evans, appeared intoxicated.
Evans and Goodson were taken to the hospital for heavy intoxication, treated, and returned to police custody.
Evans was charged with possession of a weapon by persons prohibited, as he has been convicted for a felony. He was also given a motor summons for driving while intoxicated.
Goodson and Evans were each charged with unlawful possession of a handgun, possession of hollow-point ammunition, unlawful possession of a handgun and possession of less than 50 grams of marijuana.
Hollow-point ammunition is pricier and causes more damage to targets than standard ammunition. It is not allowed to be carried in public under New Jersey law, TMZ reported.
The two were transported to the Morris County Jail. Bail for Evans was $150,000, and Goodson's was $50,000.
The Jets signed Goodson to a three-year, $6.9 million contract in March. The team has already cut two players this month for possession of marijuana: Claude Davis, defensive end; and Cliff Harris, cornerback. Each was cut the day after getting caught.
The Jets said they are aware of the report and are gathering information, the NY Daily News reported.