Trust Dr Du, he knows a great deal about thermodynamics.
You got the first formula wrong, it is dS= dQ /T.

This was the original definition of entropy in classical thermodynamics.
Now as to what happens when we add heat to something, note that Dr Du also said
Furthermore it only holds for reversible changes

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This is very important. You should always note the conditions that apply to any statement in Physics.
Of course this presents a quandrary because there are very few natural processes that take place at constant temperature.
One such is change of state ie melting or boiling when the heat added/removed is the latent heat.
In this case the entropy of the change can be calculated from the formula.
More generally when the heat input goes to changing the temperature the formula cannot be used directly and other methods must be used to calculate the entropy. Changes that include temperature change are always irreversible.
Entropy cannot be measured directly in the laboratory.