Jenkins was released by the New York Giants on Friday for making an inappropriate and offensive comment to a fan on Twitter, calling him a "retard." He was claimed off waivers by the Saints on Monday.
"It felt amazing," Jenkins said of joining the Saints. "I'm just happy to be here, man. I like the way they work around here. I love Coach [Sean Payton], how he coaches guys. I'm just ready to play football.
"I'm blessed. I got an opportunity to be on a winning team."
The Giants said it was Jenkins' refusal to admit that what he did was wrong that led to his release.
After explaining last week that his comment was "slang" and "just part of my culture," Jenkins on Wednesday was asked if he stood by his choice of words.
"I mean, I never stood by it," Jenkins said. "I mean, you know, they twisted it. I said 'I'm sorry' to whoever I offended.
"You know, the way I was raised, we got different cultures and different terms that you use. If I was to go out there and say, 'Hey, you went retarded on the football field,' would you look at it as me talking about somebody disabled? Or would you look at me saying, 'Oh, he did some crazy stuff out there? Some amazing stuff?' It's just different terms, and I feel like people just took it the wrong way. Once again, like I did five days ago ... I'll do it again: I apologize if I offended anybody or anybody with those disability problems. I didn't mean it that way."
Jenkins, known as "Jackrabbit," said he was not upset by the Giants' decision to release him.
"Business is business," Jenkins said. "You can't control what they do in the front office. You just go in each and every day and go to work. And anybody around that facility can tell you that 'Jackrabbit' always showed up on time -- never late -- and did what he had to do."
The 31-year-old Jenkins, now in his eighth NFL season, spent the past four with the Giants, including his 2016 Pro Bowl campaign. He started every game this season before his release and is tied for seventh in the NFL with four interceptions.
Jenkins said he didn't know if his release by the Giants was about just this incident.
"I never had any problems over there other than this one," Jenkins said. "But I was just focused on playing football."
Jenkins did have some turbulent moments in the past, though. He was suspended during the 2017 season for knowingly failing to return from the bye week on time. He criticized the pass rush earlier this season -- noting that he can't cover opponents for eight seconds -- and was critical of defensive coordinator James Bettcher in recent weeks for not using him to shadow other teams' No. 1 receivers.
However, the Giants still raved about Jenkins this season, mentioning him as a team leader for the younger cornerbacks.
Meanwhile, the Saints (11-3) felt comfortable enough to add Jenkins to their locker room mix during a critical stretch of the season. Payton gave no indication Wednesday of whether Jenkins will play Sunday at the Tennessee Titans or if the Saints expect him to compete for a starting role.
"We value the player, especially the position. It's always a difficult position to find," Payton said. "He's a guy that I can recall going through the evaluation process when he was a free agent [in 2016]. We've played the Giants twice since then. But he gives us versatility, and I think that's important here down the stretch."
Said Jenkins: "I'm locked in, baby."
Payton said the Saints "absolutely" had enough time to do proper homework on Jenkins because they had done some of it in the past and because the NFL is a "small world."
One person who provided some insight and endorsed Jenkins was No. 2 cornerback Eli Apple, whom the Saints acquired from the Giants in a trade last season. Jenkins and Apple, who struggled in Weeks 13 and 14 before bouncing back in Monday's 34-7 win over the Indianapolis Colts, could wind up competing for a starting job.
Still, Apple said Jenkins is a "great addition" who is smart and savvy with great instincts and ball skills.
The Saints will have to pay Jenkins $1.2 million over the final two games of this season, and the veteran is under contract for $11.25 million in 2020 if the Saints keep him.
Swearinger, 28, is with his third team of the season after starting four games with the Arizona Cardinals and three with the Oakland Raiders. He started a total of 31 games with the Washington Redskins from 2017 to '2018.
While with the Redskins last season, Swearinger made some critical comments about Saints receiver Michael Thomas, calling him a "clown" for doing "little petty stuff," such as hitting defensive backs in the leg after the whistle.
The Saints have had some recent injury issues at safety, with starting strong safety Vonn Bell being sidelined by a knee injury last week and rookie backup C.J. Gardner-Johnson suffering a concussion on Monday.
Gardner-Johnson was in attendance at Wednesday's practice with a red noncontact jersey, so it's possible that he could return by Sunday.