I can speak primarily to social psychology, but I assume the timeline is consistent across areas. Coordinator/Manager positions begin popping up mid/late winter, early spring. The big SP conference is in February and I remember last year (when I was looking for a position) that some labs/faculty were holding interviews at the conference. I have seen others post various forums for job listings for RA/Manager positions in psychology broadly, I cannot remember any of them off of the top of my head. My greatest resource was the job list serv on SPSP: https://my.spsp.org/Careers/Job-Search (refine position type to Research). Most of these will be social focused, but as there is a lot of overlap in social and cog, I'm sure some things that are more in line with your specific interests will show up on there.
To share my experience, I probably applied to 20+ manager/coordinator positions, and sent out another 20+ cold emails to PIs whose work I was interested in, asking if they were looking for a lab manager. I got a lot of responses from the cold emails, but they were all no's. I know of others who have gotten positions this way, so I don't discourage against it by any means. Most of the positions that I found through the SPSP listserv. I had 4 interviews at 3 different institutions (none of which were my home institution and were half way or on the other side of the country). I received one offer which I took, but it is actually at a business school, not in a psych department (though all of the research I am involved in is based in social psychology). I'm really happy that I ended up where I did because it has immensely shaped my research interests and the path that I'll be taking in applications this year.
I encourage you to stay diligent and not be overly concerned about finding a position with someone who does exactly what you want to do. Also 100% be open to moving, and moving far, if it is feasible for you. I would also recommend taking a 2 year positions over a 1 year position if possible (ESPECIALLY if you have to move out of state). I intended to only stay in my position for a year but the timeline/stress would have been ridiculous and thus I pushed my applications back a year. When you think about moving and starting a new job sometime in July-September (typically) and adjusting to this then immediately also having to begin the application process... It's too much. Also, this only gives you like 3-5 months of experience in said job to list on your applications.
Feel free to shoot me a message if you have any questions. I know how stressful this can be and I'm happy to share any insight that I can.