Don't worry, you're not going crazy! This Walt Disney Recycled Animation Series YouTube video shows the scenes back-to-back, so you can the scene in one movie and then it cuts (pretty seamlessly) to the same scene in another movie.
In Disney's defense, a lot of the scenes are dancing, happy scenes and since most Disney movies are feel-good flicks, there's inherently going to be a lot of crossover.
Given how tough hand-drawn animation is (especially from the earlier movies - brand new drawings for every single scene), we can understand why they'd use the same template if they've already spent all of that time creating the original.
According to Mental Floss, "In the early days, Disney animators frequently recycled footage from older animated films. They would simply draw over existing animation cells to make new scenes with characters with similar designs, characteristics, and movements."
And you know what? We're not mad at it because we love all things Disney and it was all their work to begin with. When the trade-off of sitting through one scene you might have seen before is getting to enjoy a brand new awesome movie, that's fine by us.
Still, when you watch it all strung together, it's kind of like...