It is great that you are reaching out towards meeting your goal of gaining meaningful employment. What Jess334 mentioned about trying different options is great advise. Working in partnership with job agencies when I was a Community Mental Health Nurse, I know it can really depend on the actual Employment Consultant (EC) that you are assigned to as to whether you have a good experience or not. Every relationship between job seeker and employment consultant is unique and I have seen some of these EC's go the extra mile and become a persons main support. Other times this is sadly not the case.
I would agree with jess334 and think that giving DES a go is a great option. Like with any problem, it may take a few tries to find the right solution. For example, if someone has hurt their back or has a neck injury, they may try physio, chiro, massage, osteopath, acupuncture, pain clinic, medication etc. etc.... the options are endless but generally people who are successful in anything, have kept trying. As a job seeker with an "invisible" disability you have already tried a standard employment agency, you are now thinking of trying a DES service, you might join a public speaking group like ToastMasters to help with you cold calling anxiety, you might ask a friend to mock interview you so you are better prepared once you get your foot in the door, you might ask health professional to help you with anxiety and depression symptoms specific to job seeking etc.
Applying for jobs is tough at the best of times never mind when you have anxiety and depression symptoms. Confidence can play a huge part in finding the job that is right for you. Positive self talk can be one way to help the way your thinking happens if done regularly. Saying things to yourself before you start a job seeking activity such as "I am worthy of getting a great job" "I know I can do this" "There is a good job for me, I just haven't found it yet, but I will" This sounds simple but these statements are really important to tell yourself and to believe. It is through repetition we learn and teaching ourselves confidence requires regular reminders.
You have a great deal of insight into finding your solution already and with perseverance, you will secure employment that is right for you.
Wishing you the best possible outcome,