Through Twitter advanced search, you can only locate Tweets and it can not extract the Tweets data. Apart from this, APIs have also their limitations as they do not provide Tweets more than 3200.
Only the solution for this, use any Twitter management tool like Trackmyhashtag.
I also use this tool to track hashtag performance and fetch old Tweets for research & analysis. It is a great tool for Twitter tracking. It provides raw Twitter data and useful analytics on a beautiful analytical dashboard.
You can visit the website and the page "Old Tweets". Then click the button "Request Data" and you'll see a form which you need to fill with your search term.
Within an hour, you'll receive your Tweets data.
I hope it helps!