Firstly, an HSA account is only owned by one person.
If your husband has qualifying HDHP family coverage, he can contribute up to $6750 plus $1000 catch-up. He can't contribute an additional $1000 catch-up for you because the account is in his name only.
However, if you have no insurance other than your husband's qualifying family insurance, then you are deemed to have qualifying HDHP family coverage and can open an HSA in your name. (If you can't use the bank your husband's company uses, you can shop around for another bank, try to get the lowest maintenance fees you can find.)
Then if you also have an HSA, your family maximum is still $6750 plus catch-up, but you each get a separate catch-up provision. The $6750 can be divided any way you want but the $1000 catch up is specific to each person.
So, he could contribute $7750 (employer plus employee) and you could contribute $1000 to an account in your own name by depositing after-tax dollars. Or he could contribute $5000 ($2400 employer plus $1600 employee plus $1000 catch up) and you could contribute $3750 ($2750 left over of the family maximum plus your $1000 catch up).
His employee contributions by payroll deduction are captured on his W-2 and counted as "employer" contributions (he agrees to a salary reduction and the employer contributes the money.) When the program asks you for additional contributions, only enter your out of pocket contributions not made via payroll deduction (yours, plus his if he makes any contributions that way.)
Also in turbotax, make sure you enter that you have 2 separate HSA accounts in different names and be observant to where you are entering your numbers. You will say that you had qualifying family insurance even if it was only through your spouse. On your final tax return you will have two copies of form 8889. (If you see any references to internal worksheets, form 8889-T is for the taxpayer who is named first/top on the form, and form 8889-S is the spouse's form.)