Doing two tasks at once, such as singing and playing the guitar just takes a bit of practice and determination, and no doubt you will accomplish that in time. Once you get it down, you will forget you had the problem to begin with.
The last time I tried to incorporate an added task while playing the guitar ended in a well remembered failure.
At the time I had been repeatedly watching an SRV tape for several weeks, trying to learn some things. One day, that inner brilliant light bulb came on, and I had a genius idea.
I decided why not kill two birds with one stone and decided to practice some SRV riffs, (while plugged in), and while walking on my treadmill.
Impending disaster some would think (except me.........just a brilliant idea, one of my best).
About 20 minutes into it, I fumbled my pick and instinctively turned to grab at it and was immediately flung off the treadmill. Brilliant idea, eh?
I still have a neck nick on the back of the strat. that reminds to not play the guitar while on things that are moving. Thank goodness I learned that before I started playing a J45 (the damage would have been less forgiving).