It took me several months, but I finally secured a position a few months ago. In my experience, these jobs were extremely competitive, and I had to apply to so many before actually getting one. I relocated for the position and am getting paid a good amount. I think I had more success getting grad school interviews than I did getting research position interviews. The competitiveness and availability of these jobs probably depends a lot on what area of psych you're in as some research areas are going to have a higher need for a paid research assistant/lab manager than others. I found that many postings were looking for people with coding/programming experience, so that skill will also have some bearing on the amount of jobs available to you.
While I'm in Social Psych (so my experience might be somewhat different from yours), here are some of my tips for getting a research job.
1. Don't delay. If an application doesn't have a due date, it's better to get an application in asap. I missed out on a number of good jobs by waiting too long to apply. And even if an application has a due date, the job posting could still get taken down early if the researchers find an applicant they're happy with.
2. Emphasize different skills than you might in a grad school application. For grad school applications, you're usually trying to convince the faculty that you're a brilliant, creative mind when it comes to research and that you're capable of coming up with your own ideas and lines of research. Some of this is still important to emphasize, but a big part of many research jobs is more administrative tasks. I wouldn't always talk too much about administrative skills/experiences like scheduling meetings, keeping organized, administering studies, and writing in my grad school applications, but I think these skills are more relevant for these positions. Many times, these positions are created for an already existing research project. Usually, they need more help executing on their research plan than they need help developing new research.
3. Tailor your cover letter to each individual application. You probably won't find many jobs dealing with your exact research interests, but usually it's possible to make at least a few connections between your interests/experiences and the advertised project. Are you both interested in the same target group? Have you used research methods that are relevant to this study? Do you see the work that they're doing as having important implications? Having a personalized cover letter with stuff like this shows the people hiring that you're interested enough in their project and helps your application be less forgettable. Also, based on the job description, some skills that you usually wouldn't mention might be important to highlight (e.g., if the job requires a lot of writing then I would highlight my writing experiences). I think this is the main reason I got my current job. While I was very unfamiliar with the methods used by my bosses, they could tell I was passionate about the population they were interested in.