Godwin, 32, joined the Rams at the start of last season having previously enjoyed a successful Super League career with Castleford, Wigan, Salford and Bradford.
He suffered a broken arm against Featherstone in June, although could make a return in the end of season play-offs.
Former Wakefield academy player Wright, 22, has made 22 appearances for the Rams since arriving at the club with coach Glenn Morrison ahead of the 2013 season.
Morrison said: “Wagga is a very experienced player who has played a lot in the Super League and also at international level.
“We all see what he does on the field but I really value what he does off the field too.
“He’s capable of coaching some of the young lads and is an invaluable asset at training.
“He’s been unfortunate to have suffered a broken arm this season, and while there’s a chance he could play again in 2014, I’m already looking at him for a big year in 2015”
“Wrighty has had a bit of a tough year with injuries, which forced him down the pecking order at stages of the season.
“But he’s worked hard in training to get back into the squad and has reaped the rewards of that in recent weeks.
“We saw how valuable he is to us last year. When he was in the squad at the start of the season we were in good form, but when he got injured our form dipped.
“We’re glad to have him for 2015.”