In English, as in many other languages, there are idiomatic expressions that do not adhere to any strict rule. Certainly, Standard English demands that one use the form of a verb appropriate to the tense that is needed. But, as far as which preposition to use with an idiomatic expression--an expression that has a meaning different from the literal meaning of the individual words--the choice is based more upon usage and denotation rather than any rule of grammar.
So, regarding the above-mentioned sentence, She fell to a state of sadness," it is better to write She fell into a state of sadness mainly because this is the structure of a common idiom used to convey that someone has become despondent. Of course, the idiom employs into because it suggests the submerging of the person's feelings in this condition of sadness whereas to denotes only arrival at, not inside.
As a further note, the word despair is often used rather than sadness (She fell into a state of despair.)
This statement is a little tricky, because it is probably technically correct based on definitions but does not seem quite right in terms of common usage. A quick survey of the grammar-check sites reveals no errors in this statement; however, the two words which cause some doubt are "fell to."
The most applicable definition of fall (the present tense of fell, of course) is "to pass into a particular state, condition, or situation." It is used in phrases such as "fell silent" or "fell in love."
The definition of "into" is "expressing movement or action with the result that someone or something becomes enclosed or surrounded by something else."
In common usage, "to" expresses movement toward a place, while "into" indicates movement toward the inside of a place. If sadness can be considered a place (and figuratively speaking it can), then "into" might be a better grammatical choice than "to." Even the sound of this sentence improves when into is substituted for to; it just sounds more familiar, somehow.
To answer your question specifically, your sentence is probably not incorrect, but this sentence is perhaps stronger and more correct: She fell into a state of sadness.