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If that's the case you should just buy two tires for the same axle. But I can't really see a worn tire being so different than a new tire that it would cause that problem. I would think even low air pressure in one tire would cause the same issue.
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On all wheel drive vehicles they have clutches in the transfer case and with 4 mismatched tires the clutch is constantly applying and unapplying.
GM wants all 4 tires to be within 1" or less on each other in circumference. So many times you need to replace all 4 tires when a tire is beyond repair.
At one time a lot of tire shops did tire shaving to get the tires round. This was back in the fifties and sixties. I bought an old tire shaver back in the 80's, we sold a lot of tires to guys racing showroom stock. We would shave the tires down to 3/32 or less. The fast guys would have them shaved to 1/32.
Look for a tire shop that had been around since the 50's.