Yes, TurboTax is going to automatically apply the last-month contribution limit adjustment for any taxpayer who is 55 or over and who has HDHP coverage (i.e., you do not have to do anything). So TurboTax will apply your spouse's portion of the Self-only HDHP limit to May-July even though your spouse did not have qualifying HDHP coverage in those months.
In terms of your example, please note that the additional contribution limit for 55 and over is calculated separately from the month-by-month contribution limit calculation so you should not add the $1,000 or any part thereof to the monthly calculation as you did.
First TurboTax calculates the spouse's annual contribution limit to derive to share of the Family contribution limit for those months in which both spouses have Family HDHP coverage. The spouse's annual HSA contribution limit can be found on form 8889-S on line 3B.
Then TurboTax will calculate the primary taxpayer's annual HSA contribution limit, but remove the amount of the Family limit used by the spouse from the primary taxpayer on line 5C of form 8889-T.
Then the additional contributions for both spouses (assuming that both taxpayers had HSAs - NOTE did both you and your spouse have HSAs all year? It's not clear from your question).
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