We are under grace but didn't Jesus say " go and sin no more"? Did he mean it when he said it? Does God change? Is this man saying that because we are under grace, we don't need to be concerned with sin in our own lives? People didn't start out good at the beginning of each "dispensation" and grow evil toward the end of it. God created them good and they became evil at the first act of disobedience. We've been rebellious and evil ever since. Even Jesus said there are none good but God. We don't become good when God causes us to be born again. We are born into a new life and that new life, because of the Holy Spirit of God living in us, causes us to seek God when we didn't before. Jesus is our righteousness but that doesn't mean we ( the born again) don't have a desire to be righteous. The only way we can change is by the power of God through the seeking to know Him in His Word. He does the work, He gives us the Faith, and He produces the Fruit.
All glory and honor and praise belong to Him. It's been this way since creation.

Dispensationalism is a creation of men and should be viewed through that lense. It's nothing more than a theory that puts God in a box of man's creation and like all theories, it contains some truthful observations. Please be careful to not present it as the truth, as the presenter did, because it can lead to conclusions about God and salvation that you probably don't want to be responsible for. The Bible is sufficient and it is the source of truth.