This is called "racking" and can be caused by binding due to a carriage being adjusted too tight on the rail, rail misalignment, parallel port signals being too low in voltage, or erroneous settings in Mach3.
If the mechanical issues are not corrected, changing software settings and swapping motors will not likely correct anything. Keep in mind that the bearing carriages need to roll effortlessly but not have excessive free-play. Having them a bit too loose is ok for troubleshooting your problem though. You can readjust them once you find and correct the problem.
To adjust the rail parallelism, loosen all bolts in one of the rails (the other rail becomes a reference rail) and move the gantry to one end of travel. Snug the end bolt enough to prevent the rail from moving, but will allow the rail to pivot a little. Move the gantry to the other end of travel and snug the nearest bolt a little. This should get it close to parallel, but not perfect. Move the gantry to mid travel and snug the nearest bolt. Now move the gantry along the rail and feel for tight spots. If you find any, loosen and re-snug the the nearest bolt until the gantry rolls easily along the rails end to end. Tighten all bolts and recheck for any tight spots again.
The rail may need shims under one or both ends to get the rail parallel to the other rail in the vertical (Z) axis if the frame is not perfectly true. Shims may be needed in other places if tightening a bolt creates a tight spot. This can be a repetitive effort. This procedure uses the gantry itself to calibrate the spacing of the rails. It's very difficult to align the rails with any other measurement method that I know of due to the close tolerances involved.
At higher feed rates your motors will have less torque available to overcome any binding that's present in the rail system. Binding (drag) can be bad enough to cause the pinion gear to climb out of the rack teeth. (Been there done that) Lubricating the racks can help the situation but won't cure it. Adding more tension does about the same. Neither of these are the real solution if binding is present.
When you are satisfied that the gantry moves freely and the bearing pre-loads are good, reattach the lead screws or pinion drives and run the machine to see if you still have the racking problem. If you do, then there will be other causes to investigated.
CarveOne
http://www.carveonecncwoodcraft.com