First, to carry on to a NG+, you need to be at the start main menu and select New Game. From there you can import your already completed game. It sounds like what you've one is simply load an old save from the beginning of your last game.
After you have that new game started, you'll go through all the beginning stuff again. Then by the time you gain control of Shepard for the first time (in the cockpit of the Normandy talking to Joker and Kaidan, after Nihlus left the room), you can go into your in-game menu, and in options, you can change the difficulty setting. Make sure you select "Apply settings" then make a manual game save.
You have up until the end of Eden Prime to change the difficulty setting though. Some people have reported problems with getting the difficulty setting to stick. Keep checking in your menu while on EP to see if it's still set to Hardcore. If for some reason it switched back to Easy or Normal while on EP, just follow the same steps as above, making sure to apply settings and make a save.