I'm pretty sure Disney will help with moving. They will usually even take care of your bags for you, so all you have to do is head over to the resort and check in. If your room isn't ready your options are the same as if you were already checked-in, except you don't have access to your room. You can explore the resort, get some food in the food court or restaurant, shopping, you can even go to the parks or downtown Disney, etc.
And, for room only reservations that have a package after the room only, you can even get your park tickets, for the package, early. Not sure if it's at the concierge desk or at guest relations at the parks, but you just tell them your reservation details and they will issue you your tickets. You can also add an extra day to the tickets, if you haven't already added it with the package. That way the extra day ticket will only cost about $3 per ticket, which of course will cover the room only day.
As I mentioned above, you can get your package tickets prior to check-in, so no need for a separate full price single day ticket. You can inlcude an extra day on the package tickets, or even add it once the tickets are issued to you, to cover for the room only day. Cost would only be $3 per ticket.