Thank you for your thoughtful question and concern. I think there are three ways to view this that may be of help. First, the teaching job sounds like a good fit and revitalization sounds right. So the plan would be to make the transition to this job work, not letting it go. In other words, let’s see this as a shift toward you moving on and finding a satisfactory solution for your beloved cats.
Second, I would consider the possibility of taking the cats with you and making arrangements for their care in the country you are going to. There are several features of this perspective that I think you may find useful. While you are teaching you may be able to care directly for the cats yourself, thus extending the time with them while enhancing your own career and life choices. Then, while you are traveling, you would be able to arrange for care by either a trusted person or a care facility. This way you get the best of both worlds. You extend the time with your cats and have the direct responsibility for arranging their care, while at the same time taking care of yourself.
Finally, if you do choose to leave them, as you go I would invest in finding them a caring, loving home. Interview people who would want to care for them and find folks who have the interest and ability to extend the love you have given them.