Is it normal to start your master's study without an advisor?
Well this is more like an advice rather than an answer:
In the end all of it depends on the funding you might receive. If the advisors are sure to be available with the funding next semester, I would say you don't need to hurry to find a new research area. I assume you are interested and made some preparations for your current field. It would be reckless to throw away the work you have done so far.
However if the funding is not so sure for the next semester, I would say you go with the one that supplies you funding. In the end, you can always turn back to your old questions. Researcher is very much like a qualified worker, you are paid not only for the results you've obtained, but mostly you have the capacity to treat and obtain the result or rather they pay you because it is you who has the capacity to treat and obtain the result, so one should not take research subjects at a personal level, at least not too much.
However I have no idea how US works, so you are on your own concerning the feasiblity of my advice.