Yes, this is the correct process (because it makes the HSA custodian's paperwork correct), and no, there's not a simpler way. But read on...
Of course, as you point out, the money isn't there yet since you have been spending it. So you will have to come up with the amount that was spent out of the HSA and ask the HSA custodian to accept your Mistaken Distributions (you fill out a form and send them a check). Note that you can probably aggregate the Mistaken Distributions into one large amount if the HSA lets you.
Note that the HSA custodian does not have to accept your request, so be nice to them. If they accept, they will send you a corrected 1099-SA. But since this form will have "zero" for the distributions, it may not be handled well by TurboTax (might fail e-filing), so be prepared to leave it out if necessary, and put a note in your tax file why you did so.
Note: If your employer made contributions to your HSA, then your employer can (but may not want to) reach out to the HSA administrator to recall the funds that they contributed to your HSA. Your employer should have known that you were not eligible to receive HSA contributions, so the IRS provides that the employer, in this case, can withdraw the funds (if they will).
If you contributed the funds to the HSA, then, yes, you will have to ask for the money back. If you go through the HSA interview, once you indicate that you did not have HDHP coverage (because of the HRA conflicting coverage), TurboTax will announce that all of your contributions are in excess, so even though your contributions aren't technically in "excess" but should have never been contributed at all, you can ask the HSA custodian for a return of excess contributions. They likely have a form on their website so you can do this online. But if your employer is taking some contributions back, then ask only for the excess that you contributed (otherwise, ask for all of it).
Be sure to do all this before April 15th (unless you extend your return, in which you have to get it done before October 15th).
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