If you recently worked on an axle, I would check first it didn't pop out of engagement in the diff. One loose axle would explain these symptoms.

Frankly, I thought you had front driveshaft splines go bad. I am not sure your gloved-hand test puts enough load to really test. You can inspect them by removing the front guibo and sliding the axle out.

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David.X5
2001 X5 4.4i Sport
SOLD! at 160k miles