Bashful came into the language in its modern sense in the mid-16th century, a modification of a French word, abash, plus a suffix. Shakespeare uses abashed once, bashfulness once, and bashful eight times, usually associated with words that demonstrate the sense of modesty. "He burns with bashful shame" the poet says of the reluctant lover in Venus and Adonis (line 49). And Poins says to Bardolph, "Come... you bashful fool, must you be blushing?" (Henry IV Part 2, II.ii.72).
But something different seems to be happening in Henry VI Part 3 (I.i.39).
Warwick tells the other nobles: "The bloody parliament shall this be called, Unless Plantagenet, Duke of York, be king,And bashful Henry deposed, whose cowardiceHath made us by-words to our enemies". Bashful Henry? Yes, he is portrayed as a sensitive man, but the modern sense doesn't quite suit the association with cowardice. This seems to be more the sense encountered in such other contemporary usages as a bashful army.
It means "easily intimidated, readily daunted".