In the VA system, I had to specifically ask for a therapist who treats DID. So each institution's ethics on who they treat and see might differ. Some mental health specialists are obliged to refer out when their expertise is limited and cannot fully benefit or treat those with more complex disorders. The APA has statements on that, which is why, as a consumer, it might be best to save you time and seek treatment that is tailored to your needs - for both a good personality fit between you and your therapist along with the therapist's credentials and treatment specialties.

If you are unclear on what works best for you, you can always try finding a therapist who is trained in things beyond CBT, such as those who are also trained with EMDR, brainspotting, internal family systems therapy, CBT specifically for trauma, trauma-informed psychotherapy, feminist-approaches in psychotherapy (there are some who offer these kinds of approaches as well as those who cater to the LGBTQ+ communities), racial trauma and cultural competence in psychotherapy (which is especially helpful if you are a minority or part of the BIPOC community or part of the AAHNPI/AAPI/Asian community), humanistic approaches, and other modalities.

You have a right as a patient to voice when some treatment isn't working for you or doesn't seem to be fitting in with what you're looking for. You also have a right to request a different therapist if the current one assigned to you is too triggering or doesn't seem like a good personality fit. Some people look for different therapeutic dynamics, so that's important. It differs for everyone. So, what may work for some people may not work for others.

It depends on what you're looking for.

You might be looking for a long-term therapist, but times might be tough right now to find one. So, perhaps finding a short-term therapist in the interim might be one solution. You can always be up front and honest with the therapist by stating that you intend on finding a long-term therapist in the future, but you are willing to try short-term therapy for now.

Here's an article that the APA published regarding cases outside the therapists' skill set: What should you do if a case is outside your skill set?

And here's another article that the APA published regarding ethical pitfalls, including practicing outside your area of expertise (see #7): 10 ways practitioners can avoid frequent ethical pitfalls

The views on DID and dissociation differ, so some therapists may argue that anyone trained with minimal trauma training would be competent for DID, CPTSD, and/or PTSD with a comorbid dissociative disorder. But this may not be the case, and this may cause more harm than benefit unless the therapy is short-term and specified to one treatment goal, such as learning CBT or DBT for a short period of time (3 years or less). For those who seek treatment with identifying and working with alternate personalities, that requires someone who is specifically trained in trauma and dissociation, including internal family systems therapies and any alternative therapies. There are controversies on this topic as well as other ethical topics, but what matters is how it relates to what patients need and want in therapy. A patient who just wants to work on coping skills or certain symptoms, regardless of their diagnoses, may only seek those kinds of therapists. But a patient who wishes to seek treatment for complex disorders, such as DID or CPTSD, then those require specialists.

Perhaps write a list of what you're looking for in a therapist for your specific symptoms. If you don't know, then write that down, too.

Write a summary of the questions you'd like to ask your therapist on your first few sessions, which can address your symptoms.

Write down anything you wish to discuss with your therapist on your first few visits in order to get to know your therapist better. Your therapist will likely have questions for you, too. But most therapists will ask you if you have questions for them toward the end of their initial assessment and/or intake.

Sometimes there's an intake person who asks you questions before assigning you to a therapist. It depends on where you go, and where you feel comfortable going.

If you do a search online, you might find a more specific resource to help you. You're likely going to receive a lot of different answers here on this forum or on any social media site. But if you do a google search for what you're specifically looking for, you might find the information that is best tailored to your needs.