Deductions & credits

any contributions made by you or your employer show up on your w-2 in box 12w

you had a family HDHP for August through December so you can contribute either through your employer or personally a pro-rata portion for the year 7750 times 5 divided by 12 or about $3229. Since you were covered by a HDHP on 12/1/2023, you can use the last-month rule LMR), to make a full year's contribution.  There is a catch, however. using the  LMR requires you to maintain a Family HDHP for all of 2024. if you don't some of the contribution for 2023 is treated as an excess and becomes taxable income.

 

from IRS PUB 969

Last-month rule. Under the last-month rule, if you are an eligible individual on the first day of the last month of your tax year,  you are considered an eligible individual for the entire year. you are treated as having the same  HDHP coverage for the entire year. If contributions were made to your HSA based on the last-month rule, you must remain an eligible individual during the testing period. For the last-month rule, the testing period begins with the last month of your tax year and ends on the last day of the 12th month following that month December 1, 2023, through December 31, 2024. 
If you fail to remain an eligible individual during the testing period, for reasons other than death or disability,  you will have to include in income the total contributions made to your HSA that wouldn’t have been made
except for the last-month rule. You include this amount in your income in the year in which you fail to be an eligible individual. 
individual. This amount is also subject to a 10% additional
tax. The income and additional tax are calculated on Form
8889, Part III.