Mostly yes.
This is a somewhat complicated situation, because the song you describe is composed of several different elements which have separate copyright protections:
- The sheet music of the original was public domain, and nothing you do can change that, so it is still public domain. Anyone can make their own song based on this sheet music without paying any royalties (to you or anyone else), provided that they do not copy any protectable elements from your original melody or lyrics. Anyone can record their own performance of this sheet music, again without paying royalties. Anyone can do both of the above (rearranging the melody and then performing it).
- The lyrics, assuming they are reasonably original, would be copyrighted by the lyricist (who may be the same person as the composer).
- The hook melody, assuming it is reasonably original, would be copyrighted by the composer. If the melody is an extremely common cliché of the musical genre in question, then this copyright might not be worth very much, or possibly it might even fail to meet the threshold of originality (in which case no copyright exists).
- The song as a whole is probably subject to copyright as a distinct work, but even if it is not, the lyric and melody copyrights are likely adequate to protect the song regardless.
- Each recorded performance of the song is subject to copyright protection as a separate work, owned by the performer(s), provided that the performers had all necessary licenses to make the performance. If it is a piratical performance in violation of the above copyrights, then it is (probably) excluded from copyright protection under section 103(a) (but the rightsholders of the melody and lyrics would nevertheless be able to use their copyrights to restrict the distribution of such a recording - this exclusion is intended to punish the performers who failed to get a license, not the rightsholders whose copyright was infringed).
- Once the song is out there, someone else might want to make a cover version. Under section 115, if any recordings of the song have ever been (lawfully) published in the US, then the song is subject to mechanical licensing, meaning that such a person simply pays royalties according to a fixed schedule and they automatically receive a license to make a cover, regardless of what the composer may think of such a cover version. These licenses come with a host of strings attached to them, which cannot be easily summarized here, but the short version is that the cover may only be distributed to the public for their private use (and not e.g. incorporated into some larger work such as a music video), and royalties accrue per copy distributed in this manner (or per stream, in the case of digital services like Spotify). All of these restrictions can be lifted or modified by contract between the cover artist and the original rightsholders (of course, that requires them to negotiate and reach an agreement, whereas mechanical licenses are compulsory and cannot be refused).
(1) severely constrains (2) through (4). If you want to sue based on any of those rights, you must prove that the infringing work specifically copied from the new parts of the song, and not merely from the public domain original. (5) is not affected by this, because the copyright in a recording is independent of the copyright in the sheet music (for example, you can get a copyright on a new recording of an old public domain piece of music even if it is totally unmodified). (6) creates a bit of a gray area, because cover versions are allowed to deviate somewhat from the style and presentation of the original, provided that they do not "change the basic melody or fundamental character of the work," and that would have some overlap with arrangements of the public domain version of the song. But as long as the cover version includes your lyrics or some portion of your melody, I think it's fairly unambiguous that you are owed royalties. The downside, however, is that this may motivate cover artists to avoid using your work at all, and instead to start from the public domain version. The only thing you can do to stop this from happening is to make your version of the song much more compelling than the public domain original, and hope that cover artists are motivated to build on your work instead of starting over.
