Things to consider
I have heard from some pupils that they have difficulty playing and singing at the same time. So here are some things to consider in order to play guitar and sing at the same time.
In order to sing and play well together you need to become proficient in your playing ability, chord changing needs to become automatic and fluent. For example, if you are having to concentrate on where to put your fingers for each chord change it is very difficult to also think about singing in key and the correct words plus having to concentrate on your rhythm playing.
Practice chord changing.
So the first thing to do is practice the chords until you do not have to think too much about changing. This needs to become automatic like when changing gears on a car when driving and is only achieved by lots of practice. BE PATIENT! Any skill takes time to learn.
Concentrate on your rhythm
Practice very slowly to start with and do not try to speed up until you can keep the rhythm going constantly without mistakes, then gradually speed up. If you then make mistakes slow down a little until the rhythm is consistent again.
Use a metronome to practice.
3. Try using a metronome to practice your rhythms with this will keep you in time, again start at a slow speed you can cope with. When you can do 5 or 6 bars of rhythm without a mistake make a slight increase in your metronome speed maybe 5 or 10 beats, then after 5 or 6 bars of consistent rhythm increase another 5 or 10 beats and so on.
Play chords and rhythm together.
4. Once you have managed to keep the rhythm consistent at the required tempo of the song you wish to sing then try to put the chords and rhythm together.
Use a Capo if necessary.
By placing a Capo at different frets on your guitar allows you to change the key to suit your vocal range. So if you find the song key is not right for your voice, experiment with the Capo placed at various frets until you find the right key for your voice. For more information on using a Capo link to my article https://guitarlessonscamberley.co.uk/blog/use-of-the-capo-on-the-guitar/
More information and a Capo chart can be found here: https://www.thaliacapos.com/blogs/blog/how-to-use-a-guitar-capo-chart
Play and sing.
5. Now you should know how to play guitar and sing at the same time. Try to play and sing together.