Would the English language be different without Shakespeare's influence?
Yes, the English language would most certainly be different without Shakespeare; however, the English language also wouldn't be that much different. I do not believe that Shakespeare is some English language pivot point in which the entire direction of the language changed because of the words that he put to paper. He did contribute to the English language for sure, but not enough to have the English language be unrecognizable had he not existed.
One thing that Shakespeare absolutely did do is coin new words. Over 1000 new words entered the English language because of Shakespeare. That's awesome and a big deal. I don't want to take that away from him, but Shakespeare isn't the only guy in English language history to coin words.
Another thing that Shakespeare helped do is standardize some basic grammar structures. That's a huge contribution for sure. But I also believe that grammar would have been eventually standardized anyway. So yes, English would be different without Shakespeare, but probably not vastly different.