So will you be staying at Universal but still going to Disney on certain days?

I am with you, I would do it the other way around staying at Disney and take day trips to Universal, or do a multi destination trip. I also think you get more staying at Disney.

Which hotel is cheaper to stay at Universal or Disney?

We are doing 8 days at Disney and 3 days at Universal this trip. This is our first time staying on site at Universal though, in the past we would just day trip if we went.