Do I have to calculate how much to report out of Box 1 on my son's 1098-T? It seems to contain all payments from me and his grandmother for tuition/dorm/meal?

If you claim your son as a dependent, then you are the one who claims the tuition credit.  You may claim both the money you paid and the money the grandparent paid.  Room & board (R&B) (dorm and meal) are not qualified expenses for a tuition credit.  Box 1 of the 1098-T does not R&B.  It only includes qualified expenses (tuition and fees).  You may add the cost of required books and computers, as qualified expenses. 

If box 1 includes amounts that you didn't actually pay until 2021, you will have to adjust for that. Underneath box 1, In the 1098-T screen, click on the link "What if this is not what I paid the school" under box 1. You will then be able to enter your actual amounts. This is the amount that TT will used to calculate your credit or deduction.