I have never found it difficult to book a room at any hotel. However, I have had issues with getting the specific room I want if I wait too long (e.g., can get standard view bunk bed room at Wilderness lodge, but not pool view).
The issue with booking a room only, and then adding a package is the prices go up several times a year unpredictably. So, if you lock in a rate for a room, but then want to change to a package you do not get the benefit of the original room rate. If you are changing it in order to apply a discount, then run the numbers first. I booked my current package for our trip in 3 weeks just about a year ago. Since that time, the room rates, ticket rates, and I believe dining plan cost have gone up. When the discounts were released for my dates, I found out that in order to apply the discount they would have to change my package to current pricing and then apply the discount. However, my original "Standard" pricing was cheaper than the current pricing with a discount applied. So, I just kept my original standard rate. Kinda complicated, but I'm glad I figured it out. So, it's kind of a catch 22-book early and lock in a great rate but be stuck with whatever you chose from a $$ perspective??