Employment is not a requirement to be eligible to contribute to an HSA. The requirement is that you have qualifying HDHP coverage and no disqualifying other coverage such as Medicare. Since it appears that your Medicare coverage will start in June (no matter when during the sign-up window you apply), you will be an eligible individual for only the first 5 months of 2023 and therefore your maximum HSA contribution will be limited to 5/12 of the annual limit on both the regular contribution and your catch-up contribution.
It makes no sense for you to bother with an HSA funding distribution. You can simply take an IRA distribution (no penalty since you are over age 59½) and make a deductible contribution to the HSA for 2023 (subject to the contribution limitations) to achieve the same effect, no special involvement by the IRA custodian.
