On the 8889, Part I is for Contributions and calculating the contribution limit for a particular HSA, Part II is for reporting distributions from that same HSA, and Part III is for taxpayers who used the last-month rule in the previous year but failed to stay under HDHP coverage (this does not apply to most taxpayers).
I say "a particular HSA" because an HSA belongs to an individual; hence you each own your own HSA (if you both have one - it sounds like you do from the way you said "Both my husband and I had HSA accounts"). Note that each of you have your own 8889, which TurboTax calls 8889-T (taxpayer) and 8889-S (spouse).
Distributions are therefore from one HSA or the other. You said that TurboTax did not fill in Parts II and III for your husband; this would be correct if he did not take any distributions from his HSA (whether or not you did from your HSA is quite separate). Of course, please understand that it is OK for you two to spend money for qualified medical expenses on both of you from one HSA; this is just a question of which HSA the distribution came out of.
So if your husband did have a 1099-SA because he took spent money out of his HSA but Part II is blank, then please go back to the HSA interview and see who is the owner of the HSA that had the distribution. You can do this by editing the 1099-SA (instead of "Done") and seeing which of you is checked off just above the name of the Payer.
I'm having somewhat of an opposite problem.
Looking at the forms before final submission, I noticed that while distributions are correctly indicated for my (old) HSA and our current 'family' HSA in my husband's name; the contribution amount from my husband's (our current family) HSA has been duplicated on Form 8889 bearing my name alone.
I thought maybe I'd made an error in how I entered info in TurboTax, so I went back to check - but TurboTax does seem to have the info listed correctly under HSA summary:
it shows two columns, one each for me and my spouse;
under my column, everything is 0, except a small distribution; in my husband's column, there is a larger distribution amount, as well as 'Tax-free employer contributions'
Honestly, I can't say it's an issue with TurboTax, as I wouldn't know how to fill out this form in our case -
Part 1 q1 talks about 2019 so that's my husband's plan - but I thought the rest of the questions should be about accounts in my name - given that Part 1 (like Part 2) says "If you are filing jointly and both you and your spouse each have separate HSAs, complete a separate Part I for each spouse." And yet, it's pretty nigh impossible, given how question 3 is structured ...
Hmm, looking up instructions https://www.irs.gov/instructions/i8889
How To Complete Part I
... if you, and your spouse if filing jointly, are both eligible individuals and either of you has an HDHP with family coverage, you both are treated as having only the family coverage plan. Disregard any plans with self-only coverage.
Okay, I guess that's why it's been filled out as it has.