I second (well, I guess it’s like fifth) the suggestion for Signia or Waldorf. Depending on season, rooms at each start at approximately $250 or 70,000 points (Signia) / $550 or 90,000 points (Waldorf). Even when the cash rate increases significantly, the points stay the same. That would get a standard 1 king or 2 queen with a generic view (i.e., not park-facing). As others have said, there would also be a resort fee at each. If using just points, there is no resort fee and depending on tier you might be, if you redeem 4 nights straight points, the 5th night is free.
They properties are adjacent to each other. Amenities of either are accessible to guests of the other. Signia has a very cool lazy river, whereas Waldorf has a very pretty ‘standard’ pool. They share a coach shuttle bus that makes a continuous loop with the parks. The schedule for the bus is published online by day. I believe it stops first at Signia and then at Waldorf.
Personally, I would choose Signia over Waldorf, as I have 2 young kids and I’m not expecting a relaxing stay. Signia seems more vibrant, more family-focused, but not by a lot. I would definitely choose either/or well before other Hilton hotels such as at Disney Springs.
As for availability: wherever I’ve looked, I’ve seen plenty of the standard room availability at both. This is for busier but not necessarily busiest times of year (around Christmas, Presidents’ Day, Easter). Granted, I look 2 to 10 months out, as I like to lock in a trip and then ‘forget’ about it for a while. While I am a diamond member, this does not factor into the availability that I see.