Do you think moving him to a bed would help?
No. In fact a bed would probably make things worse because he'll just be stood at the bedroom door yelling for you.
It sounds behavioural to me, not developmental. He likes sleeping with Mummy and/or Daddy so he has developed a habit of waking so he gets to sleep in Mummy's bed or with Daddy next to him.
You need to treat it more like a behaviour issue and deal with it in the way you would deal with any other behaviour you don't like. Being consistent and having form boundaries will be the key to dealing with this.
So established that he will go to sleep in his cot, in his own room, without you in the room. Then just keep on reaffirming this. Both you and your DH doing the same.
It will be really hard work initially, so expect a lot of frustration and tears from DS and very little sleep for everyone. But if you are consistent, you should be able to sort this within a week or so.
Do as you do at bedtime - say nan night, into cot, sleep time now, leave. If he screams give it a few minutes for him to settle then repeat. Say nan night, into cot, sleep time now, leave. The leaving is important, he needs to KNOW you have gone, so don't sneak out. The key is to teach him that in the night he will be going back to sleep on his own.
I would also consider a night light if you don't already have one. Just in case of scared of the dark.