Would participating in student politics as a PhD student without an advisor create a bad impression? When I had joined the program last year, I was assigned an advisor who was the one I'd wanted to work with. However, due to things not working out (because she thinks I lack the background), I have had to start looking for a new one, and that's what I was working on all summer. I'm still in the process of finding someone by working on small projects with some faculty (in order to show them I can do the kind of research they do). This kind of puts me a bit behind my cohort. In my department, most PhD students have an advisor within two quarters. While a lot of other students I spoke with told me that it is perfectly normal to be finding an advisor a bit later than year 1, I'm very much embarrassed about being at this stage. 

So anyway, my department's academic advisor just sent out an email saying she needed someone to take up the role of President of the grad student organization of the department, and I really want to go for it. In the past, I've enjoyed working with people and organizing stuff; I really enjoy talking to people and putting forward people's problems and finding solutions to them, especially the seemingly small ones that actually do make a difference. 

My question is, would faculty members consider it strange on my part if I ended up as President when I don't even have an advisor and a research project going for me? I mean, would it look like I don't have my priorities set straight? I wouldn't want to do it if there's even the slightest chance of it creating a bad impression.
It very much depends on the quality of your PhD work. If that looks good, it can be a boost to your CV. However, if your work is middling, this may appear like escapism (or perhaps a political career is what you should pursue instead of a PhD).

This is always, as @user3209815 said, your decision. What is unlikely to work well is to take up this job and getting even further behind on your PhD work.