There's a couple of parts to this.
First, asking about history, finding the real cause of issues. This is hard for a couple of reasons. One, if you are not doing too well currently, it might hurt more than help. My T always says you shouldn't dig too much in the past if the client is not stable. That might be why you end up talking a lot about current things.
Your history cannot give you a solution to any of your issues. It's not like if you know X caused Y, you know how to fix Y. Even worse, X might have happened, but for lots of mental problems there is no 1 to 1 link between past experiences and the problem. For example lots of people with personality disorders might have experienced trauma. But there's people with personality disorders who had a perfect life, and people who experienced trauma who don't have such a disorder. So it only helps to some degree to know about your past.
But regardless of what caused the issue, the fix is mostly the same. Talking about the past can bring some understanding to certain problems, but it won't fix them. It's nice to have, but it should not be the focus of your sessions, unless that's the only thing you want to use therapy for.

Second, something to work on during the time between sessions: do you mean homework? Some styles of therapy give homework often, for example cbt. Homework is often used to develop certain skills. Or it might be something like writing a journal because you experience too much between sessions to talk about everything. But, in my experience, most of the work between sessions is thinking about the last session, trying to integrate what I have learned so far. Trying to recognize things I do in therapy in the outside world, trying to remember how to calm myself during a panic attack, stuff like that. It is work, but my T doesn't tell me to do any of this. He says some people just come every week and don't spend a second thinking about it outside of sessions. Of course your results will be different then.

If you choose to search for another therapist, I suggest you tell them what you are struggling with in the first session, and then ask how they would go about working on that? If it would help you to have homework, maybe specifically mention that. If you worry that some point in their strategy might not suit you well, tell them. And if you don't like the way they'd do it, go to the next one, until you find one you like.