In Ticket To Ride, what happens if I have a ticket that can’t be completed because other players have already used that route? Is there a way to trade it in or drop it? Or am I just stuck with it?
There is no way to get rid of a destination card once you already have it.
Destination cards count as negative points against your score if you do not complete them for a reason (think of it thematically - you are playing as a company who has taken on a contract to connect two specific cities by train, and if your company doesn't finish the contract there are penalties), and this is part of the consideration you have to make when drawing 3 and keeping 2 or 3 at the start of the game, or drawing 3 and keeping 1, 2 or 3 later in the game.
You have to strategically choose which one(s) you have drawn that you think are possible to complete, taking into account how likely it is for other player's actions to block you.
Sometimes you may be able to build around a more direct route that has been blocked by other players (especially in early- to mid-game, but less likely during the end of the game), and sometimes it is even worth your while to not take the most direct route, especially where 2 or more of your destination cards include nearby locations.
Once you have accepted a route card, you are stuck with it. This is a big part of why the game does not force you to keep every route you draw, you are able to discard some of those new ones as drawn that you do not think you will be able to complete.
Just because your opponents have blocked the route you intended to use to connect your destinations does not mean you can't complete the route. The game rewards longer routes anyways, and you can complete even the shorter route cards by going all he way around the country and back again.
This only works if there is enough of the game left to allow such connections to be made. One strategy I see played often is to try for a single, long route that has alternate options to get there and accept new route cards you can easily branch off your main route to complete.
You cannot get rid of destination cards once you have kept them. Keeping 3 (instead of 2) at the start of the game is riskier because of this (more upside and more downside), as is drawing additional destination cards mid-game.
Note that almost all destinations have multiple routes to them. If the route you were planning to use is taken, you can usually take a longer route to get there.
If there is a particular route that is critical to your strategy, claim it as soon as possible. Collecting train cards for a while lets you claim better routes, but comes with the risk that other people will take those routes before you can; this is one of the big tradeoffs you have to make in the game.
Also, try not to give away what cities you are trying to connect (such as by building out from both of them and meeting in the middle) to make it harder for other players to intentionally block you.