On the ethics of continuing current project with another advisor I have been trying to work with a certain professor for a year, since the time I joined PhD. I wasn't paid by her, rather, funded by TAships. 

This past quarter, I did a project for a class I took with her. Her student, who was the TA for the course, was super super super helpful, and was basically the driving force for the project. 

The thing is, I really like this topic and want to continue working on it. Unfortunately, this professor has still not warmed up to me. She still gives me a very unclear response of whether or not she wants to take me as a student and has told me she'll tell me in a couple of weeks. Since summer is approaching, I really want to finalize an advisor ASAP, or at least someone who would supervise at least one research project for me. I am sick of not having a supervisor for research, as despite having topics that interest me, I am unable to find specific problems to work on. 

Anyway, my question is - I have given up on this professor saying yes to me at all, so I am writing to other professors; would it be ethical for me to continue my current project with whoever agrees to supervise me? I am, of course, assuming here that someone will agree to do that, but just in case that happens, is it okay for me to do this?

The project was for a class project, was not funded by this professor Iam trying to impress,but it was a very theoretical one, and the idea, the methods, everything was her student's (who was the TA for the class).
It's really not clear what you are asking. Here is what I understood from your question: You took a course from Professor X. Part of the course was a project, which you did with generous input from Professor X's TA. The course is finished, but somehow the project is not. Now you want to take this project and shop it around for supervision by one of Professor X's colleagues. You want to know whether this is OK.

This strikes me as a misguided approach on a number of levels. First, I don't quite understand how a course can be finished, when the project that was part of the course ("a course project") is still going on. (Did you decide to do some additional work on it?) 

Second, how much of the work on the project was actually your own, as opposed to the TA's? Could you have done the project on your own? Could you do a similar project on your own going forward? That is what research is, and the supervisor's role is mainly to make sure you do not take too many side tours in your research, and do not waste time re-discovering known results.

I do worry a little bit about your statement "despite having topics that interest me, I am unable to find specific problems to work on". Admittedly this is often a daunting task, but taken everything you said together, I am wondering whether you are properly prepared to even start a PhD project. My diagnosis (from reading liberally between the lines) is that Professor X shares this concern, and this is the reason why she is not ready to commit.

You've been in your PhD for one year; depending on the field that may be a rather short period for finding a thesis topic. Did you finish all your courses? How did you do? In particular, how did you do in the course taught by Professor X?

I think, if you really would like to work with Professor X, and you are interested in the topic, it would be best for you to talk this over with her, in particular addressing the question of whether she feels you are ready. Shopping the project around to Professor X's colleagues is definitely not a good idea. You did not come up with the topic, and you needed generous help from the TA to do the work. You are not going to be able to rely on the TA's input going forward.