Do I eliminate AMT by selling the shares during the calendar year of exercise or simply by selling the shares within one year after the exercise?
You eliminate the AMT by selling, gifting, or otherwise transferring ISO shares during the calendar year of ISO exercise.
Example: You exercise ISOs in January 2021 and sell the shares in December 2021.
If you sell the shares in the calendar year after the year of exercise, the spread at exercise is still part of the AMT income calculation, even if the sale occurs less than a year after the date of exercise (e.g. you exercise in November 2021 and sell the shares in May 2022).