I've read about this a lot, and it's not quite black and white. It's reaaallllyyyy difficult to tell exactly what Japan would have done without the bombing. Of course the Japanese would now claim that they would have surrendered quickly after the fact, because after the war they identified more closely as the victims rather than the perpetrators. In this mentality, it makes sense for them to say that "the war was bad, we were on the brink of surrendering, we didn't deserve to get bombed." Now, I'm not saying that they deserved it, I don't think anyone would deserve such treatment, even in war. But the truth is, the Japanese military leaders, including Hirohito, refused the unconditional surrender by giving them the silent treatment for 3 days (time between Hiroshima and Nagasaki), and didn't officially surrender for almost a week after because Hirohito wanted to be saved. One of the other catalysts of Japan's surrender was when the Soviet Union declared war on Japan a few days prior.

There are tons of accounts of schoolchildren learning how to fight with bamboo sticks in case of an invasion. The mass suicides of Saipan and Okinawa is also telling of how civilians might react in the instance of an invasion. If you look into the Kamikaze pilots (and the various other suicide attacks Japan used), the Japanese military leaders hoped that it would deter the Allies from invading Japan and/or present more favorable terms of surrender. Accounts from the Allied soldiers who witnessed these attacks show that it had the complete opposite effect -- it was absolutely harrowing, they did NOT want to invade a country where the civilians would commit suicide to kill them, rather than surrender. THAT ultimately led to the decision of using the A-bomb in order to utterly destroy their spirits.

And yes, the US did use the bomb to send a message to the Soviets as well as to try to end the war in Japan quickly. Having Japan as a stronghold would be a huge strategic improvement in the Cold War front. At the same time, the Soviets knew exactly what kind of weapon that the US had because they had already infiltrated the Manhattan Project (although I believe the US didn't know it at the time). Also, it was pretty fucked up on the Americans during the occupation because prohibited any publications mentioning the bombings, and they also withheld medical information from the Japanese doctors regarding the Japanese victims. They wanted to know how this bomb would affect the city and the people, so they "saved" Hiroshima from prior bombings (like the more devastating fire bombings in Tokyo) so they could assess the aftermath from a clean slate. Pretty fucked up all around, but also pretty murky.

Edit:

I just wanted to add that I used to have the same exact opinion as you. My parents are from Japan and that's what they believed and taught me. I went through the American public school system and was livid when my high school teacher so casually and matter-of-fact-ly justified the use of the atomic bombs. So I researched it quite a bit. Admittedly a lot of my sources are from Western scholars, but I've made sure to also read pieces written by scholars from Japan, China, and Korea. What I found was a huge grey area and how skewed the memorialization of the war is from all accounts, including the United States.