How big is this lift going to be, and what weight will it carry.
The motor you have will draw more current than the drive can supply i think.
As load is applied to the motor the current requirements will go up.
Why not just use 2 relays, 1 DPDT relay for motor direction and 1 SPDT relay for motor power on / off.
If this is just a small model you are making then the drive and rail parts from an old bubblejet printer would work well as 90% of the machanics is already built for you.
I have used many of them in various projects.
Pete.