The amount listed in box 5 for my dependent daughter's scholarship is for the entire school year (fall 2019 & Spring 2020). Do I enter the entire amount of the scholarship and then for the question (any amount of scholarship not towards 2019) enter the amount that was applied to 2020?
You are correct. Enter the 1098-T exactly as printed. If the amount in box 5 exceeds box 1, the program will ask you if the excess was for 2020. Answer yes and it will ask you how much of that excess was held by the school for 2020. Enter the amount, and if there's a difference in the excess amount, and the amount held for the 2020 fall semester, it will be taxable income to the student.
You will need to write down the amount too, because I don't think the program will "remember" that amount for the 2020 taxes.