In my opinion, internship is similar to apprenticeship which is primarily a training in a particular trade or some professional activity for which some academic qualification stands already acquired by the intern. That's why the remuneration paid during internship is called "stipend" and not wages payable to a full fledged workman or employee. Here the issue complained of by the poster is his continuation beyond the period of 3 months internship in the organization with enhanced remuneration. Generally, internship does not guarantee employment on its completion. Therefore, if there was any internship agreement, the poster has to verify it. In the absence of any specific orders as to either the successful completion of the internship by the poster or as to his appointment in the organization, I don't think that any complaint in this regard will stand legal scrutiny, when the poster had already resigned and the same having been formally accepted.
That apart, I would also add that mere continuation of internship with enhanced remuneration beyond the specified period would not, ipso facto, confer any rightful claim on deemed regular employment subsequent to the individual's resignation.