Parallax, predicted by the heliocentric model, was only observed in 1806. Until then, its apparent absence was a strong argument against heliocentrism. (We just didn't know that stars are so far away.)
DNA structure is not a very good analogue because here we already had the X-ray image, and the question was to find a molecular structure that matched the data.
Also, the discovery of DNA fulfilled a prediction by Darwin's theory of evolution: that there had to be a mechanism by which traits were passed on to the next generation. It's something that made sense within the paradigm of the time, and it fit the observations.
Heliocentrism made sense on some level, but not on another, and it didn't fit the observations of the time. Only when Kepler made a model that fit observations, and Newton's theory of gravity explained why it had to be that way, did heliocentrism made scientific sense and was geocentrism obviously wrong.
DNA structure is not a very good analogue because here we already had the X-ray image, and the question was to find a molecular structure that matched the data.