If you retrieve your reservations on the new website only to see them disappear, try this: point your mouse over "My Disney Experience" and click on "My Reservations." Near the top right you should see your name. To the right of that should be an arrow. If you click the arrow, a drop-down box will appear and you can select "My family". This will allow you to view reservations in which you are not the primary guest, or if (for some odd reason) you were not logged into your profile when you booked the reservation originally.
Also, if you are not seeing the "Done" button when logging into the "New website" for the first time, and are only seeing the option to cancel, try resetting your viewing preferences. If you reduce the size of the page, you will likely see the done button. This generally happens when viewing the website with Windows 8. If you are unsure how to change your viewing preferences, you can also try accessing the site through an alternate browser like Google Chrome or Mozilla Firefox.
If you are having difficulty retrieving your dining reservations, there are a few things to check. The first is your phone number. Dining reservations are linked via your confirmation number, name and phone number. If you do not have a phone number in your account profile, that could affect your ability to retrieve reservations. Check to be sure that there is a phone number listed in the profile and it is the same number used to book your dining reservations. If you booked by calling, there are some reservations that will unable to be retrieved online. Otherwise, if you booked the dining reservation online, and still cannot link your dining reservation, you may want to call the dining specialists at 407-WDW-DINE to confirm the phone number, name and reservation number, as well as your email address. All three points of information must match in order to link the reservation to your profile.
The mydisneyreservation.com link seems to be working (for the time being) for online check-in). If you are within 60 days of your arrival, you should also see a link to online check-in on the new website when viewing your reservation details. A good rule of thumb is if you spend more than 5 minutes trying to complete online check in, it's better to check in through the traditional route when you arrive at the resort.
I hope this helps. If you continue to have challenges with the new site (or old website) you can give the Disney Internet Help Desk a call at 407-939-7765.