TL;DR: There is one principle in asking forgiveness. If the other side decides to grant it, they will do it under the premise that, from now on, things will change. You have to convince them that this incident will not repeat. This takes time which you didn't give them.
By pushing them into situations which you are interested in (adopting you as grad student, helping you in the conference/project, etc.), you are essentially pressurising "your" prof into doing what you want; when that didn't happen, or the response - which might have been irritable, we do not know - was not quite as gentle, you fired off an official complaint.
Essentially, from the perspective of the prof, you turned part of your anxiety into theirs (no one likes to be getting a letter from the dean with a complaint with potential disciplinary consequences, not even if they are about to retire). In addition, from their perspective, you imposed your agenda on them with the outlook of negative consequences for them if they didn't comply. It does not matter that the complaint was about their tone - maybe their tone was a response to your pressuring, we do not know. So, in their mind, there will be a link between them not complying with you and your complaint, whether it was intended by you or not.
Now, having anxiety disorder may have caused this whole avalanche to be not under your control and people nowadays have a higher degree of understanding for this.
A genuine apology can be accepted when it's clear that you have understood the problem; not only of complaining to the department, but in fact, of creating pressure on the prof to further your agendas.
By immediately connecting your apology with the LOR request, you nullify the effect of the apology: you signal that you still pursue single-mindedly your interests and the apology appears just to be wallpapering over this. I quite understand that this may not be the case, or a consequence of your anxiety; but this is how it comes across. This is a "Yes, but" apology, in the sense that you "get done with" the apology, put it behind you and continue with business-as-usual. Such an apology fails to convince that things will not continue as in the past and a situation like that will not repeat.
After an apology, you need to give it a lot of time to heal. As with a damaged limb, it takes time before you can use it again, if at all.
[Note that I have only talked about the human factor and not at all about the legal ramifications of the matter which creates considerable additional potential complexities.]
What to do
I think in the present case, you lost the lead and I do not see much point to pursue it further. I am not sure in the particular situation whether you had any serious lead in the first place, as your favourite prof clearly did not want to work with you. You have tried to convince them for a bit, but overpersistence is impressive only in movies. In real life, it creates the opposite effect, especially in our times where people have far more opportunities than time to pursue them. Why should they embark on a collaboration that they can expect a lot of difficulties and pressures from?
In sum:
- Important for you is to understand the other person.
- They have an agenda, too, and you need to respect it.
- Do not turn your anxiety into theirs.
- Do not turn your problems/agendas into theirs.
- An apology is message that you accept that you did wrong, not a ticket for a second round. Maybe there will be one at some point, maybe not.
Reduce the frequency drastically by which you ask people to do stuff for you (You have not only pressured your prof, but also the department with your repeated requests - "continuously trying" - to take the grievance down). Only send every (4th, 5th, 6th, depends on frequency) mail you want to send. Note that your department has already put you on arms length with the cease and desist letter - you need to convince them that you will not proceed with your current strategy; the only way to do so is by "doing less", especially "pressuring less", not "more". Do not try to inundate them with explanations how you understood the problem now.
Accept that you will have to select a different person to work with if you do not wish (or are unable to) to change departments for a fresh start.
And, apart from above short-term advice, of course, as you probably will already do so, getting professional support to help you overcome anxiety will be the medium/long-term strategy to prepare you to avoid such situations in the future. Good luck!