That verse was part of what God told Israel under the old covenant.

(Deuteronomy 7:12-16 NIV) If you pay attention to these laws and are careful to follow them, then the LORD your God will keep his covenant of love with you, as he swore to your forefathers. {13} He will love you and bless you and increase your numbers. He will bless the fruit of your womb, the crops of your land--your grain, new wine and oil--the calves of your herds and the lambs of your flocks in the land that he swore to your forefathers to give you. {14} You will be blessed more than any other people; none of your men or women will be childless, nor any of your livestock without young. {15} The LORD will keep you free from every disease. He will not inflict on you the horrible diseases you knew in Egypt, but he will inflict them on all who hate you. {16} You must destroy all the peoples the LORD your God gives over to you. Do not look on them with pity and do not serve their gods, for that will be a snare to you.

It's totally false that Christians don't get the same diseases as everyone else. My late first wife had kidney disease. She died when she was 30. I'm a diabetic. Christians get every disease known to mankind.

Yes, God has the power to heal. If you're sick, go to the doctor. God uses the doctors and medicines of this world to heal people. Sometimes God heals people directly but it's impossible to know for sure if that's happened.