Dodge failed to punt out of bounds on the final play in a December meltdown against the Philadelphia Eagles, a blunder that resulted in DeSean Jackson returning the punt 65 yards for a game-winning touchdown. The loss ultimately cost the Giants, who tied the Eagles with a 10-6 record, the NFC East title and accompanying playoff berth.
After his release, Dodge said Weatherford actually offered him what he considered the best advice.
"Steve made a good point because I was so afraid of getting cut and I've never really been cut in my entire life. This was the first time, and I'm still breathing, I'm still alive, my house wasn't burned down," Dodge told The Star-Ledger. "I survived that, and now I can just go carefree, let talent take over and not worry about getting cut or something like that.
"I learned you can go through the craziest year of your life in the NFL, come out and still be breathing, still be fine and still be confident."
"I saw his growth and maturity and focus, which was illustrated most by a game here at home when he was booed by our fans and he punted the ball 63 yards, and after that, I didn't hear much," Coughlin said.
Dodge said Coughlin told him Weatherford's ability to kick directionally was pivotal in his decision. Dodge, 24, admitted he must improve at that and said he was surprised it didn't lead to his release sooner.
"That's another reason why I can't be upset they brought in competition," Dodge told The Star-Ledger. "I just think back to last year when, shoot, thought I was getting cut every week. For some reason, they stuck with me. Going through that, it helped me learn (fearing a release) isn't the best way to go about it. You'll end up falling on suicide watch.
"But the confidence they showed in me, always having my back ... they believed in me probably more than I believed in myself last year, which is pretty awesome when the best franchise in the world gives you that kind of support."
Weatherford, a seven-year veteran who spent the past two years with the New York Jets, averaged 46.2 yards with a 43.9 net during the preseason. Dodge averaged 46.4 yards with a 34.9 net.
If Dodge isn't claimed off waivers Sunday, he said he'll pursue workouts with some teams -- perhaps even the Eagles.
"Wouldn't that be something?" he told The Star-Ledger with a laugh.
Rosenfels hasn't played for three weeks because of strep throat, an infection that also led to back problems.
The Giants also terminated the contracts of six veterans: wide receiver Michael Clayton, tight end Daniel Coats, guard Ikechuku Ndukwe, center Chris White, defensive tackle Gabe Watson and kicker Rhys Lloyd, who handled the kicking duties the past three games with Lawrence Tynes battling an injury to his right leg.
Also waived were quarterback Ryan Perrilloux; running backs Andre Brown and Charles Scott; tight end Christian Hopkins; wide receiver Darius Reynaud; offensive tackles Jamon Meredith and Jarriel King; center Jim Cordle; defensive ends Ayanaga Okpokowuruk, Craig Marshall, Alex Hall, Justin Trattou and Dwayne Hendricks; and defensive backs Joe Burnett, David Sims and Jerrard Tarrant.
Safety Brian Jackson (hip) was waived/injured.
Coughlin, who's trying to return the Giants to the playoffs after missing out the past two seasons, said this is a tough day to be a coach.
"Some of the guys are really hurt," he said. "Some of them were shocked. This is a day when their expectations -- you teach them to aim high and they pretty much all aimed high. There was some reality there, too. When you tell them that this time around, unfortunately, it's not going to work, it's a difficult thing. It's part of the business. I've done it for a lot of years. We do things as a team here. It doesn't get any easier."
The Associated Press contributed to this report.