I gave up n the Mighty Mouse a while ago. I recall when Apple used to advertise how indestructible their mice are.
I hope Apple gets their act together and designs a great mouse, because there really are no great alternatives out there.
I really doubt this is a dirt clogging issue. There is actually only 1 "click" in the Mighty Mouse (i.e. one physical button that clicks when you press down on the mouse). If you hear it click, then it has in fact clicked. The mouse knows which "button" you are depressing by touch-sensitive panels on either side of the mouse. It is likely that these sensors are getting confused or are "weak."
The first step would be to throw in a fresh pair of batteries (even if the mouse doesn't say that the batteries are weak).
My next suggestion would be to try dragging while making sure that only your left-click is engaged (hold your left-click finger firmly on the left-click side of the mouse while holding your right-click finger far away from the mouse).
If this is a wireless mouse, and if you still have an issue, delete the mouse from your software, and then set it up again as a new bt device.
If you still have an issue, brig it to an Apple Store (or call Apple), complain, and hope they will give you one that works. Then decide whether it is worth keeping or whether you should sell it on eBay.