As someone who went through the American Legion when I applied for both treatment and compensation for my service-connected disabilities upon leaving AD I have some
experience with this even though my situation wasn't quite what you're dealing with since I didn't retire from AD. However, membership isn't required to obtain assistance and the various veterans service organizations (VSOs) can assist you with applying for disability and helping you navigate the VA system, which may include gaining access to treatment through the VA. To do so will require you to sign a limited power of attorney before any VSO can assist you with said system. The Priority Group
category you fall into, that's based upon your financial resources and/or disability rating for problems that are determined to be service-connected (or aggravated by military service), will determine if you can be seen/treated at VA facilities for the problem you're experiencing. The VSOs can provide much
more information about the requirements and application process as I can only speak for my own situation.