Yeah, Shazam really does not work for that. I tried singing melodies in to find a piece of music I just heard and could not identify. That does not work at all.
Better suited is something like this:
But I am not sure how well it works. Whenever I tried to find a title from aural memory, it did not work very well to find something. But it does give you some known melodies based on what you put in. And that might be a good starting point to see, how many melodies are similar to yours and how similar they really are.
On the other hand, I wouldn't be to worried. Chances are that nobody will really care if you accidentally use something quite similar to something existing. That would only happen when you really land a hit or do music for a financially very successful movie. Then, somebody might get the idea to make some money out of a similarity. At least, that is what usually the prerequisites for anybody starting a aw suite.
I have been copied in one cases by the next guy who got a job, but I did not start a law suite about it, although I was pissed. I was pissed, because it was obvious that he just copied me from the last project for the same client. That couldn't have been an accident. He just used my idea cause the customer liked it or maybe even told him to make something like it. It made me angry, but it would not have been worth it to start a law suite for that.
And on the other hand, when your melodies and harmonies are REALLY simple, there are gonna be hundreds of similar works out there. It is not going to be a problem. It only becomes a problem, when you deliberately copy something less simple that is famous and try to make money from being similar to that. That might lead to somebody wanting to sue you.