Can I write new lyrics to an already existing song?

In this case you need the permission of the originator, who is in most cases represented by a publisher. The publisher will evaluate the new lyrics and give approval or not. This process can take time. As mentioned in chapter 1.1 - Which songs am I allowed to upload according to copyright law?, the original authors need to be credited truthfully in the metadata.

Without permission of the publisher and therefore the original author, you are not allowed to upload a song with new lyrics. This is illegal and will most likely lead to a copyright lawsuit.