The decision to drop the bomb is perhaps the most controversial decision of the war -- which is why it's the subject of The Most Controversial Decision by William D. Miscamble, who is my main source for this.

Ultimately, a compelling case can be made for either side about the morality of dropping Fat Man and Little Boy, but there's little disagreement that the Allies could win a land invasion of Japan like the planned Operation Downfall.

The Soviets had already broken their non-aggression treaty with Japan and were sweeping through the under-manned and poorly-equipped forces in Manchuria, and were moving towards Korea. The Empire of Japan had few resources left with which to mount a counteroffensive, their manpower reserves were low, and their major cities were already besieged by Allied aircraft -- Tokyo in particular was on the receiving end of a heavy firebombing campaign. The Japanese navy had lost supremacy in their home waters, and their air force had lost control of the skies -- not to mention Operation Starvation (the laying of sea mines around Japan's major ports) was severely affecting Japanese naval logistics, and Japan was being heavily blockaded. The Allies had just about every conceivable strategic advantage for an invasion.

So no, the bombs weren't the only way to push a Japanese surrender; they were just seen by Truman as the better way out. If they didn't work, Operation Downfall was always an option.