"I am looking for a way to obtain all tickets within a date range, from [start date] to [end date], but I see that the Zendesk API only handles a maximum of 1000 records per request. I am interested in getting all the ticket information (without events). Any recommendations?"
I recommend using the Incremental Export API instead. You'll have a max of 1000 results per page but you'll be able to use cursor based pagination to retrieve all the data that you need.