Letter Boxed, part 2: this time, solve it
Jelly, 38 bytes

iⱮⱮL&gt;2ƊƇẠƇ’:3IẠƲƇŒPU=ḢɗƝẠ$ƇFQL=ʋƇ12ịḷḢ


Try it online!

A dyadic link taking the flattened letters as left argument and the dictionary as right. Returns a list of strings of one of the shortest solutions, or a zero if there is no solution.

Full explanation to follow once it’s a bit better golfed.