We were on the Caribbean Princess, the Crown's sister ship, in June, and didn't feel too crowded at all. And our ship was at capacity. Were there lots of people? Yes. But there were still lots of times where we ran in to the same people over and over again who we recognized and said hello to.
Some of the more popular shows filled up, and so you may not be able to catch everything at the last minute, but we never felt the ship was too busy.
If we felt it was too crowded, we wouldn't have booked the Crown for April, which we have.