Pulmonary Embolism Support Group
By far the most common form of pulmonary embolism is a thromboembolism, which occurs when a blood clot, generally a venous thrombus, becomes dislodged from its site of formation and embolizes to the arterial blood supply of one of the lungs. Symptoms may include difficulty breathing, pain during breathing, and more rarely circulatory instability and death.
If you've been coughing a lot, it makes sense that you could be getting sharp pains in your chest. I mean, any sort of harsh violent motion like coughing (or vomiting, in my case) can really trigger old aches and pains and create tons of new ones.
It's impossible for anyone on here to give you a definite diagnosis. (Actually, it might not really be possible for anyone to give you a definite diagnosis... might just have to see how you respond to treatment.) What they are telling you sounds totally possible to me at least, and it doesn't sound like they've short-cut your treatment anywhere. Hang in there and see if it gets better...