There's no absolute way to know, especially as inventory changes daily. However, you can get a good idea in a couple of ways. Go to either the
DCL site or cruiselocator.com...or expedia.com. Work thru the system as if you are placing a reservation for a family of 3. In the DCL system, it will show which categories are available, in most other systems, it will give you 6-8 choices of staterooms in each available category. Repeat the process as if you are 2 adults only.
If you get a choice of 6-8 staterooms in each category, your cruise is not full (or anywhere close.) If you get a difference in availability between a booking with a kid and a booking with 2 adults, you know that cabins are available, but the kid programs are full (for the age you selected). If you only get 2-3 categories available, your cruise is getting full.
It's not accurate, but gives you an idea. Just remember that DCL continues to try to sell the cruise till the date of the sailing, and offers various incentives as the date approaches....like FL resident discounts, special prices to travel agents, specials to cast members, etc. Some of these specials don't kick in till 30 days before sailing. Captain Mickey likes his ship to fill up, even if he has to offer some incredible deals to make it happen.