Over the years I have done a fair bit of songwriting and, unless I am parodying a well-known song, I would try to write an original tune for my song. Apart from that, I have written a number of tunes (songs without words for those on iTunes!).
I am pretty self-critical, but might ask a musically knowledgeable friend or 2 to listen to my "original tune" to check that I am not plagiarising someone else's tune, even if that person has been dead for a couple of hundred years, and their tune is out of copyright: not that I do any of this for commercial gain!
It can be relatively easy to change a tune by altering a few notes or "inverting" or "reversing" it, e.g. Instead of writing a phrase like ABCEDAB you write EDCABED (inversion) or BADECBA (reversion)