If you are "covered" by a family HDHP, you can make contributions to an HSA in your own name, even if you are not the named primary insured person. Coverage is determined month to month on the first day of each month. It sounds like you and your spouse are covered for all 12 months, even though the primary insured will change. So your personal limits are $8300 each, with a combined family maximum also $8300. Meaning you can divide the $8300 limit between you in any way that you want.
Note that if you can make contributions by payroll deduction, you save more than by making after-tax contributions, because payroll deductions save the 7.65% social security and medicare tax, in addition to state and federal income taxes. But you can still make contributions in any order or combination you want since you are both "covered" for the whole year.
This answer has been verified for accuracy by an Intuit expert employee
