I'm surprised you don't have one already - I don't know exactly how it works in the USA compared to up here, but again, I will bet that the 1st doctor you see if you bring up stress as the cause of your health problems, will immediately break out the pad and start with xanax and other benzo/etc type "anti stress" pills. My advice is if you haven't been taking these, try to avoid them. My mother was put on them, and it was hell getting her off of them. While they CAN work, and in some cases, probably like yours where the doctors/health pros believe that stress is causing some major havoc such as you describe in your OP, it may be your best option unfortunately. Just do a lot of research about whatever they try and give you on your own, and for god's sake, do NOT use any more than they prescribe, try and use LESS if anything.
IMO I still think stress alone isn't likely to be responsible for your OP description of what's happening, contribute yes, but primary root cause - the odds don't favor it IMO, and I've had plenty of time in the last years to read and research a lot about similar things. Make sure they do complete blood/urinalysis panels, test anti nucleic antibody for connective tissue disease indicators, and really pay attention to your SED rate in the tests. IMO what you've posted, trouble sleeping and frequent infections is pointing RIGHT at a immune system disease or problem.
I had similar issues as you've described, + others, and was told everything in the book from 4 different GPs before finding the 5th who discovered that it was SLE/Lupus. In fact, the US/Canadian average for patients with such immunodefiecient type diseases is 2+ years and 3+ doctors before getting the correct diagnosis. I can send you a lot of info to read about such things if you wish. Also, I'm not saying you even HAVE any issue other than stress, just that based on experience of both reading and living in doctors offices/hospitals for years now, it would surprise me if that was the only problem.
I've found you really have to take personal responsibility these days, and not just count on the doctors/nurses/etc to just "figure it all out" and "get it all right" - IMO it's as much on you to try and figure out wth is wrong with yourself as it is on them, but they are the gatekeepers to the tests and equipment needed, and if you're fortunate, with a good doc, he/she will have experience and good intuition as well, and will listen closely to your descriptions of what's been happening. Krabs, be sure you have everything in point form written down when you talk to a GP, and accurately describe every problem and issue with the dates. Also keep a record of what you're told in every meeting with any medical pro as well - I record mine, but some countries that isn't allowed, but you can always just take notes.