I am currently a Ph.D. candidate in a Chemistry program (4th year beginning), my advisor is diagnosed with a serious disease and in hospital recently. He does not agree with me graduating with a PhD degree so fast and he told me to look for a new university to apply for a new position. He asked me to look for a TA position at the beginning, but after I did find one then he just changed his mind and told me he just would not have the money to cover the research. I tried to look for similar research in the current department, but there was just no similar research area. If I need to apply for a new PhD program, how should I phrase the reason and if there is anything i need to be extremely cautious? Should I look for a higher-ranked university or inverse? Thank you for reading this post and I really appreciate any recommendations.
Under the circumstances and if medically possible, the professor should take some (a lot of) responsibility for placing you somewhere or getting you done where you are. The professor is the one with the most relevant professional contacts. He might have to ask a colleague to act on his behalf, of course.
Under the circumstances, another professor and their university might be willing to forgo the usual admissions process to get you a new position. I'm assuming some professional contact of your advisor here.
Under the circumstances, your own university, through the department, has responsibility for getting you through expeditiously. Perhaps another professor can help you get done based on the work you have completed, perhaps with some outside advice.
You can and should exploit any/all of those possibilities. The responsibility lies elsewhere, not solely on your shoulders.
But to answer your question explicitly, say that your advisor is dying and can no longer supervise you and that you have a (large) body of work toward the degree already completed. But, others can say that on your behalf. For example, your department head could make an appeal to another department head (different university) explaining the situation and ask if there is someone who could "adopt" you. Perhaps you can avoid taking comprehensive exams at a new place.
But the basic idea is to try to exploit personal contact webs among the faculty at yours and other universities. They can be very powerful in such unusual cases.