I'm glueing down 3/8" engineered hardwood (5" planks) on 36" wide stair treads and the 36"X 84" hallway at the top of the stairs. I understand that no expansion area is needed for the stair treads.

The planks in the hallway will be run at a 90 degree angle to the stair treads. I will butt up the ends of the hallway planks to the top solid oak stair nose. This 7' long hallway has two doorways on one side and one doorway on the opposite side.

Because of a the framing for a built-in linen closet in the hallway (also on the wall with one doorway), I would like not to leave any expansion space on that one 7' wall and have expansion space only on the opposite wall and the 36" wall opposite the top stair nose. All of the walls are drywall.

Am I asking for a buckling problem

Engineered flooring is much more stable than solid hardwood flooring.
You will not have a problem unless the flooring gets flooded.

Why not undercut the sheetrock for movement? Engineered can move in length more so than width. Anytime you but tight can be an issue. I'll leave a small gap and fill it with a silicone based caulk.

What do the flooring manufacturer's installation instructions say about an expansion gap?