Oh he sounds vile, and I really don't know why you are still interested in him.
For a start, your children have been ill and you are ill. Although in Covid times I can understand if he's kept a distance, anyone who loved you / cared for you would have been regularly checking you're OK and asking if they could drop off some food, pick up prescriptions, just keep you company by chatting on the phone. Anything to make your life a little better at such a difficult time.
The fact that he's gone off the radar and been hard to track down tells you everything you need to know about what sort of person he is / how he feels about you. Everything.
For that alone I would dump him.
The other stuff is just the icing on his poo cake isn't it. When you get to the point that a random woman is messaging you saying he's cheated, something is badly wrong. All his actions suggest he's up to no good. He is certainly lying about something (possibly everything).
He couldn't send you any clearer signs that he's just not that bothered about you.
Then there's the gaslighting.
And he has the cheat to call you controlling!!!!! I think if you were controlling, your bf wouldn't be out with another woman and ignoring you!
Please ask yourself why you think you deserve this. Because you don't. This man is no catch. He's awful. This is not a good relationship. In a good relationship you both feel valued, respected, cared for, listened to, supported, appreciated. You share your highs and your lows, your worries and the workload. You leave each other in no doubt at all about your feelings. You don't have to analyse or second-guess what the other person means. You know you are at the centre of that person's world.
Does that describe your r'ship? And if not, don't you want to find that r'ship? You would surely be much happier single than with someone who treats you so badly, and doesn't care how that affects you.
Please please dump him, embrace being happily single. And when you're ready, if you want, look out for a r'ship with someone who will treat you well, with respect. Such people are out there. You've just got unlucky with this one - he doesn't have it in him to be a good bf.