Paulski's advice is good.
If you want a more mechanical/methodical approach.....
There are many notes that will work with a chord but there are always 3 to choose from that will definitely work (4 if it's a 7th chord). These are the 3 notes that make up the chord.
If it's an F chord for example, F, A and C will work. If you can play the chords yourself, you can work out which one of these 3 (if any) you're already singing and force yourself to try one of the other two. Perhaps forcing yourself to change every syllable that is on the same chord. Or create a simple rule for yourself such as same note, same note, different note.
When moving from one chord note to another, you can always use the note in between on the way. This is called a passing note.
If you have an F chord again and you are singing and F and then going to an A, the G which sits in between these two will work.
Again force yourself to try these out. If you are having trouble doing this vocally, maybe test out the melody on a piano or other instrument instead or get one of those friends to help you out.
Yodasdad