Don't worry. You can sing and you don't even need a bucket or a lid to carry the tune. I only heard minor pitch issues and I'm sure that was because you were playing at the same time. For recording purposes, that's not the best way to get a good track. Sure, playing live is sometimes necessary, but in this case, you'd sound better singing the vocal alone, while listening to the finished accompaniment.
It stands to reason. You don't have to listen to what you're playing on the piano at the same time as you're listening to what you're singing. Not that it isn't possible, but especially if you aren't totally familiar with the song, you are thinking of too many things at the same time to get everything just the way you want them.
What I'd do is sing the song enough that you can do it with relative ease. Once you get familiar with it, breathing in the right places, pronouncing the words, getting the phrasing down, etc, you'll sound much better. I'm sure this was just a simple recording to see if you should continue singing or not. It was not intended to be perfect.
Well, now you know you can sing in tune. But, there's more to it, than that. Like I said, finding places where you can breath without it being noticed and that usually includes getting the phrasing correct. Sometimes it's necessary to be overly correct in pronouncing some lyrics, to get the vocal to work. Sometimes hesitating a bit off the beat can also loosen up an otherwise stiff and rudimentary sounding vocal.
All of this comes from repetition. Sing the song a lot of times, but without playing the piano. Record all of the music, so all you have to do is think of the vocal. Record it and then listen to it. Critic it. Record it again, maybe changing something, to see if that is better or worse than the one before. After a while, the song will come easy to you and that's when you are getting close to being ready to record it for keeps.
You can also punch in a part, if that works best, instead of singing the whole song every time. Once you get easy parts down, you can work on the tougher parts. But, sometimes punching in can also have its problems. You are doing something other than singing again, and that can take your thoughts way from the primary goal, which is getting a good vocal track. Maybe record on another track, if you have one available. That way, you don't have to worry so much about the mechanics of recording...just singing the song.
Practice singing. Listening to your own voice can be strange, at first. We don't hear our own voices the same way as others do. When it's recorded, that's how others hear us. Get accustomed to hearing your voice on a track so you can put that strangeness out of your head.
All of this will work, with practice. You have the necessary skills. Now you need to put them all to use.