I have seen Moose all along Moose Alley and most times are in the roadway before I see them so be careful! Moose's eyes do not reflect light like a deer and being dark colored they are very hard to see at night. I have been up there at times and seen no moose at all and other times there were so many moose everyone was driving around with their hazard lights on.
As far as off road and dirt roads for SUV's - There are many unmarked roads up there, especially in Pittsburg and Milan NH. I know most of them are now covered in snow and are also snowmobile trails so be sure not to drive on those. But there are some other roads that are just not maintained that may be fun to drive through with a 4x4. There are also a lot of logging roads, especially in Milan and Errol. Ive been down many of them in Errol myself, beautiful up there and if you can get out on some of the non maintained roads its very quiet too.
Are you staying at a hotel or renting a cabin up there?