Sounds like it may (Stress MAY) be leaking at the joint in the middle and letting the water migrate within the roof structure to cause wet spots elsewhere.
Try the roll roofing if you can. That will be the best long term temporary fix. If not, tarp/plastic and sand bags will be ok. Just watch how much weight and how it is distributed.
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I do the things that I must, and really regret, are unfortunately necessary.
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