This is a very bizarre grammatical misstep. I don't think I've ever heard anything quite like it before. There would be no grammatical problem with saying "I've sat here and cried tears." That would simply employ a compound verb phrase with one verb intransitive and the other transitive (no rule against that, I suppose). But in inverting order within that basic phrase and juxtaposing the first verb ("sat," the intransitive one) with the subject and auxiliary verb (as embidied in the contraction "I've)", an implication inevitably arises that "tears" is the object of both verbs ("sat" and "cried"), as the co-ordinating conjuction "and" puts the two on equal footing, and the one closest to the object, "sat", forms a precedent for how they must be related to the object ("tears"). In addition to being non-sensical, that is agrammatical, since "sat", being an intransitive verb, can't take any object. One merely "sits"; one does not sit tears or sit anything else (and "sat" is here the past participle of "sit"). It could be grammatically correct to use the related transitive verb "set", and say "tears I've set here and cried". But that would still be non-sensical in light of the nature of tears (which are hard to set anywhere) and what is meant by crying them, and the fact that verb order would tend to suggest the singer set them somewhere before crying them, hardly feasable even if setting tears somewhere were easy. A grammatical way to express the intended meaning would be "No telling how many tears I've cried while sitting here. But to refer to tears that one has "sat and cried" just doesn't work grammatically.