Get your taxes done using TurboTax

"I can't withdraw the excess contribution (and interest earnings) for 2023, they will be considered a carryover from 2023 on to my 2024 return, is that correct." - Yes, that is correct.

"and an application date of October 2023 which means I should have stopped making HSA contributions in April 2023." - I want to make a comment here. When you file, SSA backdates your coverage only when you are beyond your FSA (full retirement age). You will have to figure out what your full retirement age is, because it is no longer 65, but increases ever so often. So, basically, if you file before your FRA, there is no backdating at all. If you file after your FRA, then your "insurance" will be backdated up to 6 months, but never before your FRA date.

When you amend your 2023 return, TurboTax will ask you if you want to withdraw the excess...just say no, because the time has passed.

"However I also will have excess contribution in 2024 that I have not withdraw yet. I withdrew the contributions from July - Aug 2024, but my contributions from Jan - June 2024 were also not allowed, how do I handle that?  "

Here is a secret - although all the texts say that you can't contribute to the HSA once you are under Medicare, that isn't how it works in practice. Instead, the limit is based on an annual calculation - and TurboTax doesn't know when you made the contributions.

So, let's pretend that your annual HSA contribution limit is $3,000 (it's not) and you contributed $100 a month, even during months when you had Medicare. Since your total HSA contributions is only $1,200, and since $1,200 is less than $3,000, none of your contributions in this example are in excess.

So to answer your several implied questions, we would have to know the date of your FRA (see above), the type of your HDHP coverage, the number of months you had the HDHP coverage and no conflicting Medicare, and how much you contributed over the course of the year, even during months in which you were covered by Medicare.