"ONLY by believing that God sent His Only Son to die on the cross for my sins."
Actually that does absolutely nothing. It is simply accepting an idea to be true. Even the demons believe in such a manner.
Christian faith is our whole response to God in everything we do, not simply believing ideas to be true.
"You can do good works and be really good and still get into heaven."
True. Jesus even gives an example what it looks like.
And behold, a lawyer stood up to put him to the test, saying, "Teacher, what must I do to inherit eternal life?" He said to him, "What is written in the law? How do you read?" And he answered, "You shall love the Lord your God with all your heart, and with all your soul, and with all your strength, and with all your mind; and your neighbor as yourself." And he said to him, "You have answered CORRECTLY; do this, and you will live." But he, desiring to justify himself, said to Jesus, "And who is my neighbor?" Jesus replied, "A man was going down from Jerusalem to Jericho, and he fell among robbers, who stripped him and beat him, and departed, leaving him half dead. Now by chance a priest was going down that road; and when he saw him he passed by on the other side. So likewise a Levite, when he came to the place and saw him, passed by on the other side. But a Samaritan, as he journeyed, came to where he was; and when he saw him, he had compassion, and went to him and bound up his wounds, pouring on oil and wine; then he set him on his own beast and brought him to an inn, and took care of him. And the next day he took out two denarii and gave them to the innkeeper, saying, "Take care of him; and whatever more you spend, I will repay you when I come back.' Which of these three, do you think, proved neighbor to the man who fell among the robbers?" He said, "The one who showed mercy on him." And Jesus said to him, "Go and do likewise."
Sometimes truth does not suit your creed. It will still be true nevertheless.
JMW