Starting with the youngest player and proceeding in clockwise order,
players take turns until the game ends. On your turn, you must do one
(and only one) of the following three actions: draw Transportation
cards, claim a Route, or draw Destination Ticket cards.
https://ncdn0.daysofwonder.com/tickettoride/en/img/tt_ny_en_rules_2018.pdf
There's nothing there about having to claim new Destination Tickets. Destination Tickets are purely a matter of end-of-game scoring; they have no direct effect on the game state during play. You don't reveal what Destination Tickets you have until the end of the game, so other players have no idea whether you've completed them or not (although skilled players will be able to make deductions from your play). So if there were a rule saying you have to draw new Destination Tickets once you've fulfilled your current ones, it would rely on players being honest (and observant; if you're focused on other things, you could not notice that you've completed all your Destination Tickets, especially if it's a circuitous route). Moreover, it would punish players for finishing their Destination Tickets, and finishing Destination Tickets is a major part of the game. Game makers tend to try to avoid rules that punish players for achieving the goal of the game, reward dishonest players, or result in players that are not paying attention looking like they deliberately broke the rules.