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 2017 and sell the shares before the end of 2017.
If you sell the shares in the calendar year after the year of exercise, you are likely to incur the AMT, even if the sale occurs less than a year after the date of exercise (e.g. you exercise in November 2017 and sell the shares in May 2018).
See a related FAQ on the tax consequences of ISO exercises.