There are two separate points.
To make deductible contributions to an HSA, you must have a qualified HDHP and no other medical coverage. The FSA counts as medical coverage since it can be used to pay out of pocket expenses, so having the FSA disqualifies you from making new contributions to your HSA. You could start making new contributions to the HSA in 2022 if you are no longer covered by the FSA and you are still enrolled in a qualified HDHP. An HSA does not have to be sponsored by the employer, you can open a private HSA at any participating bank and make out of pocket contributions which will give you a tax deduction. (Or you can use your existing HSA if they will accept out of pocket contributions.)
Separately, once you own an HSA, you can use those funds for any qualified medical expenses for yourself, your spouse and your dependents, even if you are not eligible to make new contributions. It doesn't matter what kind of insurance you have when you want to spend your HSA funds.
