I'm currently in the process of writing a song. I've got a tune and I'm putting the lyrics together but I'm always concerned that any tune I think of might just be another song I've heard somewhere randomly that I don't remember hearing.
Do I just have a shitty memory or is this a problem that other people have too?
It's normal.
Especially if you're writing in a popular style, tons of musicians just use the same chord progressions over and over .
Once you start layering things on the melody, it's pretty unlikely that it will resemble anything too closely.
As long long as you're just using your own creativity, I think it's very unlikely you'll just clone a song.
If you're really worried about it, you can just change a few notes in the melody on the page in a way that you wouldn't think to sing "naturally"