It seems like it is still a bit early for you although don’t let me be the one stopping you.
I’ve seen people starting singing early with success but also people developing major and minor rhythm issues because of this.
If it breaks down, you might want to simplify it to a very basic level.
Don’t be ashamed, building this kind of auto-pilot requires simple beginning but you must build upon a SOLID base in order to progress and solidify that autopilot.
Brilliance on the basics
Make sure you play that single “strum per count/beat” well and do some humming, lalala or the same line all over again. Do it so much you cn’t do it wrong any more.
THESE are the foundations so you must make sure that your rhythm is SOLID before you think about making it faster or more complex
Record yourself
This takes courages because hearing yourself sing can cause massive cringes 
Get over that and be really critical for yourself. NOT for your singing. this will grow but your RHYTHM schould be SOLID now!
And of course, search the forum for useful posts like this!
Good luck!
btw I only had the courage to try singing and playing when I was fluent with playing songs with E and A shape barres and with a real solid “open chords” routine under my belt, being able to play without looking. No looking = less brainpower needed = more to allocate to timing and singing 