Quitting a PhD program from one university to different university due to change in research interest with the current advisor
Definitely move on.  I would talk with both chemE and mechE department heads at your current university (they may have an option for you without having to move on to another place).  In parallel or series, you can apply elsewhere.  I wouldn't even dwell on the flareup with the other professor but maybe try to make it a positive that you have some previous research.

Finally, be careful what problem you pick.  Make it tractable and likely to succeed. Not building a supercollider, not waiting for a Shuttle experiment, and not relying on questionable previous work.  Oh...and pick an older, established advisor who does not have an aggressive personality.  

At this point, you really should just find something and get the union card done.  Don't worry about discovering gravity.  Some research problems can really be done in a very short amount of time (I know Ph.D.s who wasted years on a flawed problem and then did something else for a year and wrote it up and got the sheepskin.)  Time is ticking for you personally so you need to look out for yourself and do something that works and quickly.