Leave of absence to leaving PhD program
In terms of whether it is worth it to continue to pursue a PhD, that's up to you and your own life and career goals. Based on your reasons for leaving the program, though, you may have a better chance of continuing in another program than students who leave for other reasons (such as failure to complete qualifying exams, conflicts with a supervisor, etc). You will probably need to explain your leave of absence, at least in vague terms.

Although US programs often admit students for direct PhDs, at least in engineering disciplines it is not that uncommon for students to be admitted with Masters degrees. At my institution, I just checked a couple of engineering programs and they allow at least some credits from a prior masters program to count towards the course requirements of the PhD, and they expect students with a masters degree to progress through the program more quickly. It's still more efficient by about 1 year to start as a fresh PhD student and skip a separate masters, but the masters degree is not "wasted" (whereas in some other disciplines there may be no transfer at all). Of course all of this will vary by institution.

There is little distinction between MA and MS; MA in engineering is more unusual but may occur when engineering programs build out of departments more commonly housed in the arts. Since your institution offers both, then that distinction matters more than the letters. Probably the thesis-based masters is better, but I suspect that it will matter little compared to a course-based masters plus the work you've done towards a thesis (especially if this includes publications).

If you decide to continue towards a PhD, I would suggest you begin contacting professors at other institutions that have research programs you are interested in, and simultaneously checking the posted policies for the programs at those institutions. You can also apply directly to programs you are interested in (hopefully those that have more than one professor in your chosen field; the narrower your focus the more likely you need to have an advisor in mind ahead of time). Pretty much the same process as a new student. Your former advisor may be a good resource for suggestions as far as who to contact if you are still on good terms with them (I don't see anything in your question that implies you are not).

Letters of recommendation from your former advisor and possibly others at your current institution that can explain the situation will also be helpful.