Were the Allies really backed against the type of corner that would warrant such extreme action?
They weren't the "only reasonable option," but they were an option that from the perspective of those in charge of dropping them had the most benefits, the fewest negatives or chances for going wrong. It wasn't that the US was "backed into a corner" — the Japanese were essentially defeated, but they had not surrendered, and in there lies a big distinction, because the US was not willing to walk away until the forces that led Japan to war were no longer in charge of the country, and their ability to wage future wars destroyed.
As for other options, I wrote a piece last year about several "alternatives" that were on the table at the time. The long and short of it is that there were certainly several avenues not pursued. (And we are not counting a ruinous land invasion as one of the possible "alternatives" here.) The ones I mention are:
Vary the schedule. That is, add time between the bombings, so that the full processing of Hiroshima could have been done before Nagasaki. Originally there was meant to be a week between the two; it got compressed to three days by local weather considerations. Three days is an awfully short amount of time, especially since it took about a day for Tokyo to get news of the Hiroshima bombing, and another day to get confirmation from scientists and soldiers about what had happened. So they really had about 24 hours to think and talk about it once they had gotten all of the information in about it. That's not much time for a government, especially one that was at that time essentially ruled by a committee of ten or so people, who needed to meet and come to some kind of consensus.
Demonstrate the bombs. This was floated by some scientists, and rejected by other scientists. The idea was to set off the first bomb in a way that wouldn't hurt anybody, just to show what it was. Not to drop it on a city first. Sometimes people say, "but they only had two!" but that's only if you go with the "drop them both in early August" schedule. The US would have had a third by late August, and they were projecting the ability to produce at least three per month afterwards. Plenty of time to amass quite a few before the November invasion.
Change the targets. That is, something other than cities. Or at least not cities chosen to maximally display the destruction.
Clarify the unconditional surrender requirement. We don't know if this would have changed much, but we do know that the major sticking point in the Potsdam Declaration for the Japanese was ambiguity about the state of the Emperor and the Imperial House. The US Secretary of War, Winston Churchill, and a top US military planning committee all urged Truman to clarify that the Emperor would not be tried as a war criminal and that the Imperial House would be allowed to exist. Truman deliberately opted to not clarify this, even though he knew, from intercepted intelligence, that this was a major stumbling block towards Japanese surrender.
Wait for the Soviets. The US already knew the Soviet Union was going to declare war on Japan soon, and it was expected that this would have a massive effect on the Japanese will to fight. (Truman, in his journal at Potsdam, noted that once the Soviets invaded around August 15th, "Fini Japs.") The US deliberately chose to drop the bombs as soon as they could, hoping the war would end before Stalin would get any spoils for joining in. If they had waited a few weeks, they could have gauged the effect of the Soviet's Manchurian invasion on Japanese thinking, and still had plenty of time to drop the atomic bombs if they chose to, prior to any US invasion (which was not scheduled until November 1945).
One can certainly argue whether these avenues would have brought the same ending or not — it's impossible to say, and many reasons to suspect they might not. But if the question is, "were the Americans forced to drop two atomic bombs with only three days between them, on urban cities, in early August 1945?" the answer is certainly "no."
I point out these alternatives not so much to second-guess the past, so much as I do to make it clear that there were lots of smaller "decision_s_" regarding the dropping of the atomic bomb. That is, it is not a question of "drop the bomb vs. invade," which is a relatively simple sort of choice — a false dichotomy, really. It is a question of what you do with the bomb when you use it. The dropping of two weapons on cities with almost no time between them for Japan to process what had happened, much less decide whether to decide, is not the only possibility for what could have happened. (It was not even what was originally planned, as I note in the piece — the timing got compressed due to weather considerations.)