It could be your roof is fine, basically. I bet the pitch is low and I wonder if with the snow you had some reverse hydraulic flow going on, ice damming basically. It's not like the terra cotta tiles seal to each other. A water test is best way to find out what's going on.
Ugh, I've scraped down and skimmed some popcorn ceilings before. Never fun working overhead, that has to be the worst. I scrape, mostly, from the floor with a long pole. You have to be careful with having less control, but what the heck, it all gets skimmed anyway. Right now I have to paint a ceiling like the one I did last summer. Beaded paneling between 4 x 4s that I wrapped (with boards I put a bigger bead on, then 1/4 round all around. Filling holes, caulking, then painting. I hate it. So tedious and all overhead. I'm getting too old for that. I also have to fill then sand all the seams in the wood I wrapped the 4xs with. I'm dreading going back. Getting paid and hearing "Looks great" doesn't even make it worthwhile. I have to remodel my own bedroom, including raising the ceiling. That will be similar but I'll have exposed hand hewn beams (existing ceiling joists relocated and maybe just drywall between, if walls are paneled. So it ain't over yet!
I took some beginning pictures, then no more. It's the S.O.S. as the other room I already took pictures of. Another thing is all the furniture is in the room, so moving, working around, climbing over and onto the bed... Did I already say I'm getting to old for this shhhhh... (we ask that members refrain from the use of foul language
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First two are her son's room I'm doing now (son lives in Peru). Last one is her room completed. She stripped the wall paper and had me fix the plaster cracks, skim it all, sand of coarse, and paint. Like I said, "ugh", none of this is my favorite thing to do