That’s correct, the limit on a user timeline is 3200 Tweets, and the limit on the search API is 7 days of history.
The only reliable way to get all the Tweets for an individual user via API is to use one of our commercial Gnip products.
As an alternative, if you have direct access to the account i.e. it is your own account, you can request your account archive via settings, and you will receive a zip file containing the Tweet data. This is not the same as using an API.