Mrsfuzzy - Sending big hugs to you [hugs]
Your opening post might well have been describing my oldest DS. In fact I did a double take in case someone had posted on my behalf!
My DS has ASD (would have been AS under the old criteria), he was only diagnosed earlier this year. He does also have depression and quite severe anxiety, which has gotten much worse over the past couple of years - he's barely able to leave the house, especially not alone.
It might be worth asking if you can call the GP to speak with them in advance of an appointment for DS, or as you said, writing a letter explaining your concerns.
Also take a look at this: www.wired.com/2001/12/aqtest/
It's sort of a screening test for symptoms of autism on a "friendly" site. Perhaps if DS scores highly, you could print off the results to send to the GP in order to ask for referral for diagnosis.
Just to warn you though, autism screening and support for adults is woefully inadequate these days. DS was waiting for over a year from referral to diagnosis, and we've just been told that the local autism has centre has lost most of it's funding.
Things that might help...
As others have said, getting a dog (if DS would like one) would perhaps give him a sense of purpose and a reason to go out, without having to worry about interacting with others. (We have a houseful of cats, so not really an option for us).
Pokemon Go? We discovered this a few days ago, I installed and showed DS asking him to help me. He loves it, and has been out with me several times now hunting for pokemon and getting items from the stops.
I've found a good way to get DS to open up is to ask "how are you feeling?" rather than badger him to wash/go out/do stuff. He doesn't always answer but feels able to talk to me when he's able. In the same vein, try texting/emailing DS. He might find it easier to open up without having to be "face to face" IYSWIM?
Ask DS if he could help you with something (going to supermarket, fixing something, etc) and show that you appreciate his company and his help. Give him plenty of advance notice though - if he's anything like my son, asking him to switch off mid-game will only cause him to be upset and unwilling.
Hope this helps a little. It sounds really difficult for you all 