Maximum HSA contribution

No. Your overall family limit is $7750 with prorate, you and your wife each have a $1000 limit that is individually pro-rated, but your combined limit can't be more than the family limit.  Your wife does not get an extra $4850.

Let's also start by clarifying that, if you enroll in Medicare from 3 months before to 3 months after your 65th birthday, Medicare takes effect on the 1st day of the month in which your 65th birthday occurs.  But if you delay Medicare, your enrollment date is backdated by up to 6 months.

So let's assume you turn 65 during September 2023, and your Medicare enrollment is September 1, and that your wife's Medicare enrollment will be November 1, 2024.

Lastly, we need to know if you will be keeping the marketplace family plan, or dropping the family plan and changing over to an individual plan for your wife when you enroll.  It seems like that is your plan, but you are not clear.

Assuming all the above, your personal contribution limit for 2023 is $5166 plus $666 = $5832.  

However, your wife's contribution limit is 

7750/12 x 8 = $5166 (8 months covered by a family plan)

plus 3850/12 x 4 = $1283 (4 months covered by a single plan)

plus $1000 catch up

equals $7449.

Your wife's contribution limits are also calculated from the type of coverage that she has on the first day of each month, even if the policy that covers her is in someone else's name.

The tricky step is that your maximum combined contribution is $6449, plus $1000 catchup for her and $666 catchup for you.  That's based on her maximum eligibility given that she is changing coverage mid-year.  You can split those amounts any way you like, except that the catch-up portion can only be contributed to that spouse's account.  If you contribute $5832 to your account (5166 + 666), then your wife can contribute $1617 ($6449 minus $5832 plus $1000).  Or you could contribute $666 and your wife could contribute $7449. Or almost any other combination.

Then for 2024, assuming you did drop the family coverage and convert to individual, and she is eligible for 10 months, your calculation is correct except the 2024 individual limit will be raised to $4150, so her eligible limit will be $4291.

"is this correct or can next year calculation be based on family contribution max (even so I am on Medicare?"

Your limit is based on whether you are covered by a family HDHP or a single HDPH.  When you go on Medicare, you are ineligible to contribute to the HSA, even if you keep the Marketplace coverage as well.

Your wife's limit is based on whether she is covered by a family HDHP or a single HDPH. She only has to be covered, not be the owner of the policy.  She is covered by a family plan at least through August 31, 2023, and then she will either be covered by a single plan (if you drop your coverage when you enroll in Medicare) or she will be covered by a family plan (if you keep your Marketplace family plan even after you go on Medicare.)  Her eligibility is not affected if you become ineligible, as long as she is still covered by an eligible plan and is not enrolled in Medicare.

If you do decide the keep the family HDHP Marketplace plan on top of Medicare and it covers your wife after you turn 65, her limits will need to be recalculated.  I assumed you would drop your HDHP coverage and switch to a single plan for your wife.