Because chemists learned that the atom was divisible, did Dalton's theory get any criticism or did they just adjust the theory to fit the new idea?
Dalton's atom was the most primitive idea. Keep in mind many physicists were against atomistic theory and cathode rays were not immediately recognized as electrons or a subatomic particles. The beauty of science is that it is self-correcting. This happens on daily basis still today. Nobody actually throws any theory out of the window or badly criticizes another until and unless they are bitter enemies. There are successive improvements in models which explain the experimental observations better and better.
A German company Miele has a nice logo "immer besser- always better". This is true for science. On the other hand, science is still in infancy. For example, can you define what is charge? Or What is mass? If someone comes up with an explanations, would we throw away all the previous work of physics or chemistry or hold the previous scientists in high esteem? Same thing happens with Dalton. His name will always have the due respect for proposing the clear cut idea of atoms.
Dalton's theory was about how atoms behaved: the observation of Cathode rays provided part of the explanation of why they behaved like this
The key elements of Dalton's theory were observational. He explained how the idea of atoms could explain chemical reactions and many observational facts about them. This was an important breakthrough and explained far more than previous ideas. But it didn't provide much of an explanation of why they behaved that way.
The observation of cathode rays was one of the first steps in providing the why not a fundamental challenge to Dalton's actual theory. Chemists didn't get a (mostly) complete explanation of what atoms behave the way they do or of some of the characteristic features of atoms until the discover of the neutron in the 1930s. Why, for example, do atomic weights not always follow the atomic numbers (or the pattern of chemical behaviour observed across the periodic table)?
We now understand that chemical properties are determined by the electrons in atoms and the number of electrons depends on the number of nuclear protons. But the mass also includes neutrons which occasionally alters the apparent periodic relationship between atomic mass and chemical properties.
None of this is a fundamental challenge to Dalton's ideas. All of it is a deeper explanation of why the observed properties and behaviours of atoms in Dalton's theory work as they do.
So we didn't throw Dalton's ideas out. We refined them with a more fundamental theory that explained why they behave like that.
This contrasts with other changes in our understanding of chemistry. For example, we discarded the old phlogiston theory of the explanation for combustion because observations didn't refine it but completely refuted it.