In TTR USA, does a player have to draw more Destination Tickets when they run out? (Meaning they've completed all the ones that started with.) Or can they simply continue to play by extending their longest route or blocking other players?
No. Your opponents should not know when you complete a route until the end of the game. Also, you score points for every track you lay down, so you aren't just playing to block other players or to get he longest route. Completing routes is simply one of the ways to score points, it is not "the object" of the game.
No, you are not required to draw more destination tickets. Nor, do you "run out" of destination tickets because they are kept hidden until the end of the game from players.
From the rules:
Players keep their Destination Tickets secret until the end of the game.
This hidden information can additionally be used to bluff your opponents, by making them think you have more or less points than you actually do!