Hey Trent,
One of the best Network Admins I have ever worked with had Asperger's. So I know it can be done... But it has some serious challenges.
Most web developers will have more interaction with a wider group of people than the average programmer in my opinion. Also, good HTML code is far more subjective than other types of programming. I wonder who diverted you away from coding... IMHO, software coding is a much, much better option.
Coders generally deal with tickets that deal with functional requirements, and when done right, acceptance criteria (need a new method on the xxyyzz controller that takes two parameters and makes the following changes to the customers account)... Web developers generally work from less focused requirements that if feature focused (we need a new page added to the web page to generate a request for our sales department to contact them)
Finally, whoever told you a programmer, especially a 50+ job changer, needs a degree to be a software developer? I have worked in many different aspects of IT, I program in Ruby, Python, Java, and Node, (plus if you get me drunk enough, I'll admit to Go), I have worked as a Network Admin, and now work supporting developers in a web SaaS company as a DevOps Engineer. And I have no degree.
I think your advisor needs to focus more on making you successful, and less on filling some companies whole. If I were helping you, I would take your unique attributes, and try and find you a Jr Developer position, or even better, Q/A and Testing. Software testing would also play very well to tour autistic nature.
Hope this was of some help.
Kevin Fries