Sounds like you had a spindle break the first time? It almost hard to mess up replacing a spindle and even the brake assembly if it has brakes on both axles, so depending on exactly what you mena by "flexing" it could be normal. The trailing axle (fring in the case of backing) will drag/scuff the wheel, especially the inside and may actualy make it look "tilted" which could be perceived as moving.
If you do a lot of those 90* turns it will put pressure on the spindle. Have a bearing come apart and you can quickly break a spindle in the right situation.
Best way to check is to get that wheel off the ground and check for movement. Shouldn't be any. If thee is, pull the wheel and hub and check the bearings.