I thought it was a misconception that Shakespeare added so much new words - him being a prolific and very popular writer, and with much of his corpus surviving to this day, his works get cited whenever there's a "first use" mentioned somewhere.

The fact that one of his plays is the oldest written source where a word was seen for the first time in written form doesn't mean that it wasn't in common use before that; indeed it could mean the opposite – because the spectators of his plays should be able to understand what the characters are speaking, not leave befuddled in need of translation.