Pushing is way way way easier than pulling up hill, holding it from rolling down hill, etc...my shoulders hurt after pulling a wagon load of kids around all day.
I'd rather have both arms facing the front and being able to push stuff up hill and hold it tight on the down hill.
I loved my double stroller. It had 2 seats, one in front of the other. I took it everywhere, even shopping at Walmart and to lunches. It was easy easy easy to pop it out of the back and unfold it. Then push away.
If you have a friend that has one borrow it so you can get a feel for it.
A large wagon is over $100 and you should be able to get a simple double stroller for not much more than that. I'd take the stroller any day of the week and turn down the wagon. I just hated keeping one arm in back of me and pulling that weight.