Try this as it should take you to the 'old' site where you can make your reservations...
Here are the steps I follow:
Open Disney.com
In the upper left corner are a number of small horizontal lines. I click those.
Click 'Parks & Travel'
Click 'Walt Disney World'
Scroll to the very bottom of the page and in small letters in the dark blue area is 'Full Site'
When I click that it takes me to the old site I'm familiar with. I choose 'Where to Eat' and am able to book dining reservations there.
Hope this helps!!