axles have nothing to do with causing the tranny to malfunction. you might hear a grinding, clunking or clicking noise but they will not cause the tranny not to shift. 116K is due time for plugs if they are the originals. is this a bmw specific shop? if not, you might want to check into a reliable bmw indy. sounds like you have the typical roasted X5 transmission. replacements are anywhere from $3000-$5000.

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2006 4.8iS Black/Black with custom Audison Voce/Image Dynamics sound
Production Date - 10/05