Another very important aspect of grace is that God's grace doesn't leave us as we are. Instead, through His grace, God works to help us become better people. I would define ungodliness by setting up a scenario. Suppose Jesus Christ Himself were to show up now and walk with you for a day. Those things are ungodly things in your life.
Worldliness is slightly different.
I see worldliness as an attitude that has priorities set by others around us, not by God. If our motivation is to gain fame among those around us, we are acting out of worldliness. We should live sensibly, righteously and in a godly manner. If so, then we are living righteously. Nobody does that perfectly, which is one of the reasons why Jesus had to die on the cross.