Pub 936 does not provide guidance on how to calculate the average balance of the two mortgages resulting from the sale and purchase of a primary residence. As you have discovered, strictly applying Pub 936 when one of the mortgage balances is more than $750K results in a significant reduction in the allowable mortgage deduction. Turbo Tax apparently takes this route.
However, in the absence of specific guidance, the taxpayer may use any reasonable method to determine the amount deductible as qualified residence interest. This includes using any reasonable mortgage balance averaging method and applying the $750 limit to the average balance.
The tax code and Pub 936 center around the concept of the taxpayer having one primary and one secondary residence and applying the $750K limit to the (aggregate average balance of primary residence mortgages) and the (aggregate average balance of secondary residence mortgages). Where the aggregate average includes the first mortgage, second mortgages, and refinances.
When you change your primary residence during the year so that you now have one mortgage on the sold home for part of the year and one on the new home for the remainder of the year, it is not only fair and reasonable but, in my opinion, the intent of the tax code to include both the sold home and newly purchased home mortgage monthly balances in the aggregate average balance of the primary residence mortgages. This is the method you are asking about.
When you use this method, the aggregate average balance of the primary residence is the sum of the monthly ending balances of the sold and purchase mortgage balances divided by the sum of the number of months each home was the qualified primary residence. The number of months could be less than 12. Do not zero out months when neither home qualified as your primary home.
Both homes cannot qualify as the primary residence simultaneously. If there is a period of overlap, the sold home qualification ends on day it was paid off or the day the new home became your qualified primary residence. One possible way to handle the mortgage balance for the month of the switchover is to use the daily average of the two mortgages or use the balance of the sold home just before payoff (not $0) for that month and the full month ending balance of the new mortgage for the following month.
Although I believe this a fair assessment, I am not a tax expert and this is just my interpretation of the tax code. I have tried and failed to find any official tax document or ruling that addresses this special situation. Still, I have advocated this method to some members of the community but also have advised @KSB78 in a similar situation as yours against it only because the difference in the percentage of interest deductible in the current year (87.2%) was much greater than the expected percentage in subsequent years on the new home alone (57.7%) and who needs questions from the IRS.
