Can you use three 2x8 sandwiched with 1/2" plywood between to get a 16' header
A beam made up of multiple members with plywood spacers can be done, if the fasteners are up to code.
However, that really isn't your question. You want to know if you can reduce the depth of your header while also increasing it's length. For any given condition (loading, length, beam size, material (wood or steel)), the two things that most affect the design are those two (smaller differences have larger ramifications).
Further, for a longer beam, deflection controls not strength. Deflection is many times more dependent on the depth of the beam rather than the width.
Therefore: You CAN NOT make up the difference in depth by widening the beam.
To get a more specific answer will require a site visit by a structural engineer so they can determine the loading and other site specific conditions.