It’s not clear what you are doing with the Ajax data. Perhaps it’s because that is what you are asking about — what to do with the data? It has to be stored somewhere so when the user reloads the page, the $results array can be ordered to match the order previously saved. Depending on how long the order data should persist and if the users are logged in or not, you can save the ordering in cookies (no Ajax required for this), PHP sessions, WP transients, or user meta.
When outputting the table, fetch the saved ordering data, which should be sorted to match the table data default order so the order column from saved data can be merged with the main table data, then re-sorted by order column values.
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