Use the up and down method. If the 67K shocks are original, they aren't in good shape (not to mention good initial quality). Some of the side to side is the tire sidewall flex.

 

I've got a 1987 F150 with 67000 original miles. It seems to move easily when I grab the back of the bed and shake it side to side. Pushing up and down on the bumpers seems to be fairly normal, one and half bounces before returning to normal. Could the side to side movement be caused by worn shocks?

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1987 F150 with 67000 original miles.