((((((((Rose76))))))))))))
Having been in a similar predicament at one point, I can relate to your fears as I had them myself. I hope this might help.
I can say about the SSDI.... it is a requirement to qualify for SSDI that you HAVE worked. So whomever said you have TOO much experience to qualify is not accurate. I wish I could find one of those really good links for you to read about the requirements but am coming up short.
I hated having to file for similar reasons as you mentioned but I had no choice; I would lose the roof over my head and was down to the bare minimum of food, as unemployment was ending, and then my severance and my savings were dwindling quickly.
The process was very respectful yet it can be stressful sometimes. However, it is worth it. It is set up to help you at some point return to work when you are ready, with retraining through vocational rehab or another agency, if you choose, but provides income in the interim so you can do this without hardship.
It is a lengthy process though and does not happen overnight. It is important if in a position that you cannot work due to a physical or mental disability to get the process going by filing sooner than later.
I don't know if would affect your unemployment but I know the time it took for mine and others' claims to be approved by SSDI, my unemployment ran out before the SSDI was approved. It is also not uncommon for the initial application to be denied and have to appeal which takes even more time. If even your doctors are on board by suggesting it to help you, let them, for some relief until you can figure things out and can get in a better place. Seriously consider filing.
I know it is not an easy decision but the world turned right side up for me once it was approved. The stress relief of having some income again so I could get my feet underneath, pay bills, retraining, and back to work in a positive way. The help is there, just have to ask.
Hang in there in the meantime! Hugs!