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 2019 and sell the shares in December 2019.
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 2019 and sell the shares in May 2020).
See a related FAQ on the tax consequences of ISO exercises.