Hey there! I had this same problem myself a couple years ago--wanting to create a song, but only being able to sing and write the lyrics/basic melody. I agree, it can be very frustrating.
The way I got around it was asking other people to help me out. I asked on Ichigo's Sheet Music forums and the Voice Acting Alliance forums, but I imagine you can ask around here as well. I kept my request simple, stating:
1. What I was looking for: someone who could arrange and/or play backing piano (both would be a plus), and a male singer (since the song was written to be sung by a guy, and I am neither a guy nor do I sound remotely like one). I deliberately kept my request as simple as possible, especially since this was basically one of the few times I'd posted in those forums.
2. I posted a sound clip of me singing the lyrics, along with posting the written lyrics for reference, so people would know what it actually sounded like.
3. I made it VERY clear (in a polite way) that it would be a free project for both myself and them, since in the end it was a song I wrote for fun. That meant that neither of us was going to profit monetarily from it, but I hoped that it would be a fun learning experience for all involved.
4. Oh, and last but DEFINITELY not least, I read the forum rules and kind of poked around other posts to see what kinds of posts would be acceptable, and which were likely to be ignored, or worse, deleted/locked.
I received several answers, and from those I picked an arranger/pianist and a guy singer to work with.
With the arranger, we exchanged some e-mails where she'd send me sound clips of what she was working on, and I'd approve it or be like "could you change this?" She was pleasant to communicate with and I generally liked the music she ended up arranging/playing, so I did my best not to be overly demanding, since this was after all a free project.
With the guy singer, it was basically the same. He sent me an audition, I picked him, he sent me a clip of him singing, I may have asked him to sing a part differently (or not, my memory's a little hazy about that part), and he probably did that and sent it again. Again, he was polite and I was very happy with the result of his singing.
I edited the separate pieces of audio together, and then I posted it on Tumblr!
It was overall a really good learning experience, and it was fun to make it and have a finished product.
So that's what I did! It may work for you, it may not, and that's okay. I hope that you found something helpful in this post.
And you can ask me if you have any questions, too.
For reference, the completed song (and a more complete synopsis of the process)
can be found here.