You're mostly correct. It's one of those details that we gloss over in science fiction because we're too busy telling a story to bother with details that don't contribute to the story, but yeah. The thing we're going to will have moved by the time we get there.

One detail that you forgot about, however, is the contraction of space. Granted, that may not be a thing at FTL since we wouldn't be traveling through flat space if we were going faster than light, but it may still matter. As you approach light speed space contracts from your frame of reference. I'm too lazy to do the math and get the exact figure right at the moment, but in the scenario you laid out you'd cover that 25,000 lightyears in significantly less than 25 years (from your perspective) thanks to this phenomenon.