Your Medicare coverage will start August 1, 2024, so you will be HSA eligible for only the first 7 months of 2024. Because you will not be eligible for all of 2024, you will fail to complete the testing period for a 2023 contribution made under the last-month rule. You can make an HSA contribution of up to 4-months-worth of the 2023 contribution limit without penalty. Even though you will fail to complete the testing period, you can make an HSA contribution of up to the full annual limit without it being it being an excess contribution but you'll owe a 10% additional tax with your 2024 Form 8889 on the amount that you contribute beyond 4-months-worth. (Because only excess contributions are eligible to be returned and because failure to complete the testing period does not result in an excess contribution, you are not permitted to request a return of any portion of the contribution permitted under the last-month rule to avoid the 10% extra tax.)
Your HSA contribution for 2024 can be up to 7-months-worth of the 2024 annual limit. Anything beyond that would be an excess contribution.
The amounts that you indicated correspond to the family contribution limit, suggesting that you might be married and that your spouse might be eligible to make an HSA contribution. Ignoring catch-up contributions, your combined HSA contributions are not permitted to exceed the family contribution limit.
