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Four words place D second and A third, and no two of them take the same fourth letter.
ADAPT is the only entry in ordinary use. ADAGE is familiar to most readers, while ADAYS and IDANT are the kind of words that surface in a word list and nowhere else.
Three of the four open with A and one with I, so the first box is nearly settled before you begin.
Four letters appear in the fourth box across four words, one apiece. G gives ADAGE, Y gives ADAYS, P gives ADAPT and N gives IDANT.
So the fourth box alone identifies the answer. That is unusual, and it means one well chosen guess can finish the puzzle outright.
Nowadays, in an older form. You will see it in phrases like now adays in early modern writing.
It is a term from early genetics for a grouping of chromosomal material. The science moved on but the word stayed in circulation.
ADAPT by a distance, since it is an ordinary verb. ADAGE is a noun most readers know but reach for far less often.