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I ended up adding the 1/2″ square tube down the length of the gantry. It added about 7 pounds, which isn’t terrible. After reassembling, the gantry is much more rigid.
I tested my X carriage and Z axis again by flicking the gantry front and back in the Y direction. I still got a lot of shaking from the X/Z assembly. So, I clamped a 6 foot section of 1.5″ square tube, 16ga wall, to the gantry. I pointed it straight up in the air and tested it again. It shook less than the X/Z axis. So, I have narrowed it down to the X axis plate having the shakiness.
After I added the Z axis motor and motor mount, the shaking decreased significantly. This is most likely due to the Z axis lead screw, coupled with the motor, increasing rigidity by adding strength in a different plane than that of the flat X axis plate. So, to further increase stiffness, I will add a thin (probably 14ga) steel cut out. I will offset it from the front of the aluminum X axis plate to add a second plane and increase stiffness. If this doesn’t make sense, I should have pictures if I attempt this.
I can definitely see the attraction of spending some money on a good machined Z axis, like that of a Velox Z axis. I wanted more travel, and I am stubborn, so I will keep at it!