I think you mean that you'd like a character meal outside the parks, not off site. Off site means not on Disney property and you won't find any character meals there.
As for a character meal at one of the resorts, you may find it hard to find anything. As you said, many reservations have been taken up by the 90+10-eligible guests. My advice would be to keep trying the dining line and to be open minded. Also, make sure you are calling the Group Dining number, not the general dining line. The Group Dining CMs are used to dealing with large parties and should be able to help you find something. I'd try to book something now (character or not) so you'll at least have something booked. Then, if necessary, keep trying to get something better.
Character meals at the resorts include:
Chef Mickey's (Contemporary) - Breakfast and Dinner
'Ohana (Polynesian) - Breakfast
1900 Park Fare (Grand Floridian) - Breakfast and Dinner
Cape May Cafe (Beach Club) - Breakfast
Garden Grove (Swan) - Breakfast, Saturday/Sunday only, limited characters
Other places to try, although not character meals are:
Boma(Animal Kingdom Lodge) - Breakfast and Dinner, the lack of characters could be offset by the animals you could see outside before/after your meal.
Whispering Canyon Cafe (Wilderness Lodge) - Breakfast, Lunch & Dinner - there are lots of hijinks that go on and the servers really have a good time with guests.
Good luck!