Green’s Dictionary of Slang

1. (US) to commit oneself completely.

1973		J. Godey Taking of Pelham One Two Three 101: [W]hat remained was whether or not it was workable [...] Why, then, had he gone the route?
2000		G.H.W. Bush All the Best 122: Rep Henry B. Gonzalez [Texas] went the route for the Democrats — Really Henry B. didn’t have too much but he worked his heart out.

2. to live a hedonistic life.

1956

I. Fleming Diamonds Are Forever (1958) 31: A hard, sullen woman who had ‘gone the route’ and whose body was no longer of any interest to the gang she worked for.