I suggest you find out what other DMD you could be taking and make a change from Copaxone if possible.
You’ve already been unable to take Tecfidera, and the beta interferons all also have depression as a potential side effect, so you’d have to be looking through the other drugs with your neurologist quite carefully, to avoid anything that can lower your lymphocytes (that was the reason for stopping Tecfidera wasn’t it?), or even neutrophils (when I asked whether a drug that reduced neutrophils would be a bad idea having had reduced lymphocytes, I was told it would), and anything that can cause depression.
Why not get an appointment soon with your MS nurse and see what s/he can suggest? Meanwhile, try getting an early appointment with the neurologist. You could try to arrange this by phoning the neurologists secretary and see what can be sorted out.
It sounds like Copaxone really isn’t the right drug for you, and noticing the side effect quickly is a good thing. You just need to get the right DMD for you. Having a quick look, Gilenya, Cladribine and Ocrevus can reduce white blood cell counts so they’re out. I personally would give Aubagio a wide body swerve, it isn’t very effective and has a great long list of possible side effects including neutropenia. But Tysabri doesn’t have either of these side effects listed. I don’t know whether you’d qualify for it, but it’s worth asking.
You could talk to your GP about antidepressants as well, just to make sure you are on the best drug you could be.
Best of luck.