OK. Here I will try to guide you through the data entry process. First, you need to know what you paid and when you paid it. You need to know scholarships and when they were paid.
Again if you paid in December 2018 for qualified tuition to go to school in 2019, then the IRS tells us to record that payment as a 2018 expense. It is OK to pay in December for expenses for the 1st 3 months of 2019. But if you pay in 2019. then those expenses are for next year.
You should normally enter the 1098-T via the path Deductions & Credits/Education/Expenses and Scholarships (Form 1098-T). When you get into the 1098-T data form, then enter the data exactly as written on your 1098-T. Whatever it says for Box1, then put that number in Box1 in TurboTax even it is not a correct number. TurboTax gives you an approved method to fix incorrect numbers. (Note that my version of TurboTax has a note next to an orange-colored Boxes1-6 tells that reporting varies by school.)
Here is the thing you need to do to insert the correct tuition. Look for the words, "What if this is not what child paid to this school?" (You see, they know that the Box-1 data is often wrong.) This is the situation where your own records are what will really count. You will need to know your 2018 tuition payments better than the school. And from what you write, you do. CLICK THAT LINK below the Box 1 entry on the 1098-T page, and then a box will appear that lets you insert the true number for your 2018 tuition. Be sure to enter all of the rest of the information from the 1098-T--especially Box 8 because if that Box is inadvertently missed then results might get confusing.
I recommend that after all is entered, you should then look at View/Forms (Thin black colored menu near the top of the window.) Then look for the form named 1098-T Wks for the school. Immediately below the numbered boxes in that worksheet, you should see a section titled "Reconciliation of Box 1, Payments Received for Qualified Tuition Expenses." Box B gets populated with the your corrected number for tuition. Box A will only have a number if you enter a number lower than reported by the school. The Box 5 Scholarship and Grants information gets corrected in a similar manner. There is nothing unusual about this process, and you should be fine as long as you have your records to document the true numbers.
Your school is not unique. I have prepared many of these education credits, and I find that the numbers on the 1098-Ts are more often wrong than correct. I always ask clients to bring their bank records along with the school financial records so that we can unscramble the mess and get to the truth that will hold up if ever audited.
