Copaxone is not an immunosuppressant. It's technical name is Glatiramer Acetate, and it is an "immunomodulator". It functions in a way that is not suppression of the immune system. Think of it as more of a decoy. It shifts T cells from Th1 to Th2. In other words, it's a little more sophisticated than just tamping down the immune response.
If you are in a severe current flare up, any disease-modifying treatments you begin now will not effect it. That is what courses of corticosteroids are for. Methylprednisolone (a.k.a. Solu-Medrol) administered by IV is what is typically prescribed for acute, currently active relapses. This may be something to discuss with your neurologist or MS nurse.
As for disease-modifying treatments, there are so many options available these days; multiple orals drugs, injections, infusions. It can be a little overwhelming finding one that's right in any stage of MS. But fear is not the best decision-maker. Any treatment decision is a very personal one, but having a decades-long relationship with an unpredictable illness is tough enough without thinking that my future includes the question "What if?". What if I'd fought this thing with everything at my disposal? What if I'd understood the risk was worth the benefit? The fact that I'll never have to do that have already providing immeasurable psychological benefit. I'm not rolling over for this disease. Side-effects can happen. Sometimes you have to kiss a few frogs to find the prince, but again, don't cede your decision to fear.
Flare ups are scary and its understandable why it's so nerve-wracking. Before you worry, try giving your doctor a call. They can talk with you about the symptoms you're having and options for management, which may include medication. But if you have a good doctor you like, then they'll be willing to work with you discuss all your options, including any concerns or worries you have - ultimately to find what's right for you. I'm sorry you're going through this, and hope the symptoms let up soon..
I have MS and have had it for more years than I care to tell. I have taken tecfidera but was taken off after a month due to the white cells going crazy, the nurse did not explain but I wouldn't take it. I was also on avonex but the side effects made me ill and unable to walk. Now I am on tysabri infusions and I think it is a miracle drug even though it has killed about 8 people and was taken off the market because of that then put back on since the people loved it so much. It helped me although I am now unable to walk and am in a wheel chair. I can answer anything about MS. hope this helped