Yes, to what
@Friday said, especially re prioritizing stabilization, i.e., dealing with self-harm, s/i, basics such as safety, housing, drugs/alcohol, etc., then making sure you have a good set of proven coping and distress tolerance skills, before diving into the trauma.
By proven, I mean that, if I slip up and self-harm, the trauma-focused therapy needs to be stopped or paused or can't start until I've gone a certain amount of weeks/months without self-harming. I have to prove to myself and to my T that I will be able to handle the distress of confronting the trauma head-on without resorting to seriously self-destructive faux "coping" skills.
My understanding is that all evidence-based forms of trauma-focused therapy, once you are stable, has some element of exposure to it. That includes EMDR, as well as Prolonged Exposure (PE), narrative therapy, etc. By exposure that means, at a minimum, being able to talk about the trauma without either completely dissociating, going numb, or falling apart.
That does NOT mean it's easy, or that you won't struggle with over- or under-emotionality. It just means you are exposing yourself to the memory(ies) of the trauma, without running away or becoming completely unglued. A good trauma therapist will be able to help you "titrate" (find the right balance) your emotions so that you stay within your window of tolerance.
Exposure can also mean you approach other things you avoid. I don't know if EMDR includes any homework or if you work on exposing yourself to things irl that you avoid because it reminds you of the trauma, like people, places, and things.
I recently finished 10 weeks of PE, where the protocol is to recount the traumatic memory week after week (called imaginal exposure), feeling the fear and anguish, yet pushing through it to desensitize yourself to it. NOT desensitize yourself in the sense of going numb or not caring, but for the physiological arousal and fear response to decrease. In PE you monitor your distress level to make sure that it is going down over time and not escalating. You also expose yourself to things you avoid irl, called in vivo exposure. For example, I avoid certain yoga positions because they give me flashbacks and my hypervigilence goes through the roof. My in vivo exposure was just to go to my yoga class, and work up to certain poses on my own at home.
PE was in-f*cking-tense! But it damn well worked, at least for that particular memory. I trust my T a lot, and she was very skilled in pushing me to connect with the memory and emotions when I was numbing out, yet helping me stay present when I got so caught up in the past and was starting to dissociate. It did get worse before it got better.
I think the main difference with "regular" therapy and trauma therapy is that during PE our check ins for the previous week were really about the in vivo homework. There wasn't time to meander about how my week was in general. In the 75-90 minute weekly sessions we spent about 15 minutes reviewing HW, then 30 minutes doing the imaginal exposure (brutal), then the rest of the time processing the traumatic memory. Each week it was a little different.
EMDR has an exposure component, I believe, in that you are exposing yourself to the traumatic memories you'd rather avoid. You have to be able to tell your T what happened, and what is going on for you internally. To the best of my knowledge, EMDR and PE are the only two evidence-based (meaning well-researched with proven results) trauma-focused therapies that many insurance companies, HMOs, and the VA prefer. There are other therapies out there, they just haven't been widely researched.
Hope this helped give you some insight and context! Good luck with the new T!