Though in a weakened state, the conventional thinking is that the pesky upper level northwesterly winds will finally begin to diminish on Wednesday.Conventional thinking isn't in a weakened state, but that's what the sentence says thanks to the position of the modifier. What is intended here is to say that a tropical storm has weakened. Dangling modifiers may not be as dangerous as a Category 5 hurricane, but we should still try to avoid them.
For tips on how to find and repair misplaced modifiers, try this at the Purdue University Online Writing Lab. Or consult the AP Stylebook under "dangling modifiers."