"Can I persist in staying?" Yes
"Am I right that me leaving and taking all my stuff (DH wants it gone) will be more distressing for kids?" Yes, I think it will.
"Financially could just about manage to rent a room somewhere nearby so thinking of doing this. My idea is to come in every morning to do their breakfast, and stay every night till after bedtime. This is what we have been doing the past couple of weeks since we first talked and it has worked ok ... I thought." I sounds grim for you and dishonest for the kids if they think you are still living there.
"House is in joint names, as are all savings." Then get a solicitor involved and work out what is best for the kids.
Obviously having an affair is not a good thing to do, at all. But you are the children's mum and this is your home. You and your ex/dh need to work it out together. He can't just chuck you out.
Patronsaintofglocks "... if you leave the courts will look at that as abandoning the children for another man- and you won't have any luck from that."
WatchingFromTheWings " ...change the locks." I believe that is illegal. If he does that, consult the police.
"I am the main carer for DC (have arranged work to do all school pick ups / drop offs etc) and want to stay that way."
Then they need you.
"Am I wrong to think that falling out of love with DH doesn't mean I have to spend less time with DC??" Of course you should not spend less time with our kids, but you will need to work all this out. Do not allow him to take over now because you have made a mistake in your marriage - marriage and parenting are not the same things.
ImNotDancing "my mother had an affiar, she left us with my dad. my dad is my best friend and I will always hate my mother a small bit for leaving me.
This is the bed you have made, now you need to lie in it with your OM. Why should your husband suffer even more because you broke your vows"
Why would you want the OP to leave her kids with the risk that they will hate her a little as you do your mum.
Having an affair is a horrible thing to do but it doesn't have to ruin the OP's relationship with her kids.