Consider the contribution limits being comprised of two parts, the regular part and the catch-up part. The regular part is subject to the family limit of $7,750 split between the two of you, and individual catch-up contributions.
Regarding the regular contribution limit for having family HDHP coverage, having been HSA eligible for all of 2023 you are eligible to contribute to your HSA up to the family limit of $7,750. Because your wife was covered by Medicare for the last 5 months of the year, your wife is eligible to contribute to her HSA up to 7/12 * $7,750 = $4,521 of the family limit. After determining each spouse's contribution limit separately, you must ensure that the combined contributions subject to the family contribution limit do not exceed $7,750.
As for catch-up contributions, you are eligible to contribute to your HSA up to $1,000 as a catch-up contribution. Your wife is eligible to contribute to her HSA up to 7/12 of $1,000 = $583 as a catch-up. The catch-up contributions are independent of how you split the $7,750 regular family limit.
For example if you contribute $8,000 to your HSA ($7,000 of the total family limit + $1,000 of catch-up) , your wife can contribute a maximum of $1,333 to her HSA ($750 of the total family limit plus $583 catch-up).
