The Quasit Does Not Gain Blindsight
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Its statistics are the same in each form, except for the speed changes noted.
Knowing the definition of Statistics makes this clear. From page 6 of the Monster Manual:
A monster's statistics, sometimes referred to as its stat block, provide the essential information you need to ...
Take a look at shapechange spell and notice that it specifically restricts the spellcasting trait AND class features, while the dragon's change shape does not.
Considering the way that D&D 5e rules were intended to be clear and literal (normal English) in their implementation, dragons should be able to transform into a humanoid with the spellcasting ...