Joining shapes and fills question

In Illustrator 10, I have two shapes I am trying to join. One is a T-shape with a fill, and the other is a simple box. I'm trying to put one of the transepts of the T into the side of the box. Using Pathfinder, this is, of course, trivial, and it looks fine.

However, I don't want the fill which I've applie to the T to become a fill in the box. So rather than using Pathfinder, I can use Scissors to open the path at the end of the transept and also open the path in the box where the two shapes meet, and that looks fine.

The problem occurs when I additionally try to open the path on the other end of the transept or at the bottom of the T. The fill in the T suddenly leaps out to cover the open space between the transepts and the bottom because of all these open paths.

The only solution I can come up with is to make a line with the same stroke color as the page and overlay it on top of the path segments I want to blank out. This works, but it certainly seems to be an inelegant solution, and it would of course have to be modified anytime the page color changed. Is there a better way? Thanks for any help.

Chris