DH and I both applied for Medicare A when we each turned 65, but stayed on his private health insurance through his employer. DH DID stop contributing to his HSA once he turned 65, and waited more than 6 mos after that before he applied for Medicare A. Our mistake was initially forgetting to discontinue the high deductible insurance plan for that year, so we got a little bit of $ in an account called an HRA (health reimbursement account) but then moved to a low deductible plan so fixed that.
DH is retiring at the end of the month and we both just applied for Medicare B. Because I had decided to take my SS a few years ago (I was well past FRA), my monthly premiums and IRMAA will come out of my SS. But DH is waiting until he is 70 to take his, so we will be billed by Medicare for his premiums/IRMAA (and IIRC we can pay by credit card with no service charge, and then reimburse ourselves from our HSA. Reason to use a CC is that I am chasing medallion level on Delta). So it might be worth checking if you are only in Medicare A (most likely) but yes, stop contributing to your HSA. When you do apply for part B, your card will show your effective start date for Medicare A, and then
separately your effective date for Medicare B. My DH and I filled out the paperwork (and attached his CMS L564 from his employer showing that he was covered by the private insurance, and a letter from his HR department showing his last date of coverage). So for us Part B will start Sept 1, and we now (now that my part B was FINALLY processed) have to pick our supplemental, RX and dental/eye plans (if it’s worthwhile to do those last 2- probably will).
Like you, I robbed the cradle. My DH is younger than I am!