I don't know where you're located, so this might or might not be helpful. I'm in the US and have been in therapy for years as well as been on meds. This is just my experience...
Therapists and psychiatrists basically do two different things. Ideally, you would see both and get a well-rounded kind of treatment. But if you are able to see just a psychiatrist, do it. They handle the medication side of things, which I know you know. But they do more than just talk to you for a couple minutes and then prescribe you random drugs. They will do a thorough evaluation and ask you all of your symptoms and your history and basically whatever it takes to figure out what's going on and what might be the best approach.
If you are also seeing a therapist, they will form a diagnosis based on your visits and what you talk about and they also ask for all of your symptoms and everything but they mainly just work on talking and coping skills and helping you manage. But they have very little to do with the medication side of things. They can report to the psychiatrist what they've observed and recommend things, but it should not affect the quality of care you would get from the psychiatrist if they are good at their job.
There've been times when I have seen only a therapist, which unfortunately is what's happening now because I lost my psychiatrist for insurance reasons, and there's been times when I kept up with my medication but didn't have a therapist.
So basically that was just a lot of words to say there's no reason to hold off on the psychiatrist if you can see one without a therapist. And maybe they can help you find a therapist.
Another route you could take is talking to your primary care provider and seeing if they are affiliated with a psychiatric nurse. I have done that before and it was a good experience. Just because they're a nurse does not mean that it's crap treatment. And they had a little more time to spend with their patients and it was kind of like therapy light along with medication and they have access to a lot of resources and could also get you in with a therapist if that's what you're wanting to do. Or you can just talk to your doctor and they can also prescribe you something. It'll probably be a bit more of a generic Band-Aid kind of treatment but it's better than nothing.
And I know you didn't ask, but whoever you see, you might want to ask about lithium. It's a scary sounding drug but it truly is not as bad as everything makes it seem. But it's actually the only medication that is proven to help with suicidal ideation. I was on it for years and it really is worth a try. But medication doesn't work for everyone, and not every medication will work for everyone, so no matter who you see there's probably going to be a lot of failure and a lot of experimentation but hopefully you'll find something that helps.
And also talk therapy is pretty crap unless you just need someone to vent to. But there's other forms of therapy that are actually helpful, so I wouldn't really write that off completely. Since you mentioned suicidal thoughts and everything you might want to avoid therapists who only do CBT. Look into DBT, which is a lot more suited for that. There's also EMDR, which I've gotten to try a little bit of and that seemed promising. Anyway there's other therapy options besides just talking.
