Ana M さん
This is just my interpretation, but I hope it can help you to understand a bit better.
We need to read the entire lyrics or at least the entire stanza in order to know the meaning of that particular line.
So, let's see:
The singer, most probably "he," is standing at the seashore.
From 君と口ずさんだメロディー, we can guess that "she" used to be with him, singing some song together, and is gone now somehow. Probably the beach is the very place where they used to spend some time together.
Now, he came to the beach alone, still missing her.
He sings the song they sang together, hoping to find some trait of her in the waves.
押し寄せる literally means a lot of something rushes towards someone. Here, two things are rushing towards him; one is the wave, another is the memory of their old days. Because he is looking the wave alone that he used to see with her before, a lot of memories come up to his mind just like waves rush towards him.
Then, of course he is missing her a lot. The memory just makes him cry, and the vision blurs because of tears.
(Yes, you are right, Ana M-san!)
Based on above interpretation, the line could be re-written as: