I'm not sure about the hand-foot-mouth disease but I do know that back at Christmas time my 20 month old was really sick and had cold sores in her mouth and wouldn't eat or drink anything either. What I found worked and her Dr said was at the very least use a medicine dropper and as long as she gets a few drops every so often then she wont dehydrate. I used baby oragel to numb her mouth for a bit if it was hurting her to bad and she would be ok for a bit each time. Luckily it only lasted a few days.
Hope this helps.
Good luck and take care