The fact that you are learning (or improving on) your guitar skills might a valuable tool in loosening up the old warble box and vice versa.

Many years ago, something that helped me no end was simply singing the notes I played whilst learning various scales. Eventually that turned into singing the notes I played in simple solos and finally I was in the position where I could pop a finger on the fretboard and sing the note before I'd actually played it. It also allowed me to immediately reach for and play a note I was singing, which is very handy when writing melody.

You don't have to have a great sounding voice, you will know if you are way off key if you play music as you've said, slightly out of key you can get away with if you do it confidently. Hell, Dylan's been getting away with it for years!

Remember, most people sing flat, so sing flatly with a sharp voice and all will even out. :-)