Other than getting the delaminated tape off, there's no need to remove the texture. Simply apply the new tape, feathering the mud as you apply subsequent coats. This is where the damp sponge comes in handy, it will wipe out some of the mud build up around the edges to eliminate ridges.

The key is to not use too much mud while taping and top coating. You only need to use enough to adhere the tape, sufficiently coat it, and fill any void. Anything beyond that is just more sanding later. If you're careful enough, you may not need to sand at all, just using the damp sponge and careful knife strokes will suffice.