Both AOPA and the AME (aviation medical examiner) who advised you regarding HIMS and your status, were correct.
Do not violate the terms of your military contract. If you do, it will trigger significant additional problems for you, if it leads to a medical discharge. Do what the military has directed you to do.
You have indicated that you do have a valid condition: this is the most important concern, above FAA medicals or other issues. We're talking about your health. If the medication is helping, then you need to recognize that this may be more important than other considerations, and if it has proven to control your anxiety and give you relief, then it really doesn't matter if you were ordered to take it or volunteered. It's what's in use to treat your condition and if it's working, that is what is important.
With regard to the FAA, it is not just the medication, but the underlying condition which the medication is used to treat. The FAA considers both on an individual basis. The use of psychotropic drugs and medications do come with conditions; they generally lead to deferrals, but with treatment and on a program of testing and observation, it is possible to get medical certification. Anything you do to complicate that will only make the process more expensive or difficult later. That can include being denied a FAA medical, or a military discharge for the condition. In the case of the latter, were you to vary from your military obligations and contract, would signal the FAA that your condition had already been determined disqualifying by the military; there's no reason to complicate the matter.
You can certainly speak to or attend a physician outside the military for consultation while maintaining your military treatment. While a flight surgeon may not be available to you within the military, aviation medical professionals outside the military are available to you, and there are AME's and programs that work to help people with special issuance, HIMS, etc, that will guide you through certification.
You've already been given the general information, several times it sounds like, and the AOPA representatives who explained the situation to you may not have told you what you want to hear, but appear to have given you correct data.
On a separate note, don't attempt to give up your medication to get a FAA medical, until you've got the professional guidance, treatment, and assistance to replace the medication with something else. If it's what's giving you the relief you need, focus on that: it's what's most important, regardless of how much you want to fly. Health first, everything else a distant second.