"I look forward to [something]" or "I look forward to doing [something]" is a polite way of expressing interest. You will often find it at the end of an email, e.g. "I look forward to meeting with you tomorrow." You can also say "I'm looking forward to [something]" but this is more often used in speech, not writing, in my opinion. In your particular case, I think you have got the usage wrong; you can still use that phrase, but it probably would not sound very natural. I would recommend something like "I would like to listen to new songs by you" or "I wish you would make more new songs." Finally your second paragraph has a grammatical mistake near the end; I'll let you figure it out. :)

'I'm looking forward to hearing your new songs again' sounds better somehow, even though 'listening' is gramatically correct note: you always 'listen to' something, never 'listen' something the english verb 'listen' requires 'to' after it unlike spanish

Okay thank you so much Aleksandra!! You have solved my doubts :D Thanks.

Okay so you could say "I am looking forward to hearing your new songs" if he was about to release a new album for example. In this situation you can say "I'd love to hear some new songs from you".

The situation is that my favourite singer hasn't come out new songs for a long time. So, I would like to say to him that I really want to listen to new songs again.