Just agreeing with Anne, DD is free.
The EMH in the evening is very new, but they had EMHs in the morning for a long time (it stopped for a while and then started up). Your MAW contact should be able to find out for you. If you can't find out before, you should be able to ask that at GKTW and if they don't know, I'd suggest just asking at Guest Services in the park. You can find out from the DIS site which parks are which days.
But, since MAW families get to go to the front of lines, you may find that you can get everything done in the regular park hours.