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Mechanical strain provides a knob for controlling the magnetization of the magnetostrictive-free layer of magnetic tunnel junctions (MTJs), with many applications for energy-efficient memory and computing. This requires integrating materials with high magnetostriction coefficient into MTJs, while still preserving the CoFeB-MgO tunnel barrier for high tunnel magnetoresistance (TMR). One way to accomplish this is to replace the CoFeB free layer of the MTJ with an exchange-coupled bilayer of CoFeB and a highly magnetostrictive ferromagnet like Galfenol (FeGa). Here, FeGa, a thermally stable magnetostrictive material, is integrated into CoFeB-based MTJs. We show that engineering a thin layer of CoFeB and FeGa provides a means of controlling the magnetic properties and switching field in FeGa-based MTJs, and that the exchange-coupled FeGa-CoFeB layer can be used as both a free layer and a fixed layer in the MTJ stack with TMR as high as 100%.