Some of the replies you’re getting are nuts OP, I’m sorry.
You’re struggling with difficult conditions and very young twins. The very least a supportive grandparent could do is to respect your boundaries at this time.
It sounds like she’s absolutely not respecting them. Pretending not to be ill is incredibly shit of her, and would in no way make me feel confident that she wouldn’t also lie about a fading or oncoming cold sore. People can be as blasé as they like about cold sores, but they’re very dangerous for babies and beyond that they’re a horrible life long problem to have.
That’s not even getting into the fact that you can transfer the herpes infection into any other soft tissue, like nose, eyes and vulva. Obviously most children are going to be putting their hands in their mouths and touching the rest of their body without stopping to wash their hands if they do catch a cold sore of someone - it’s hard enough to do safely as an adult which is why so many people catch the cold sore virus! It’s also notable that men/boys seem to have more resistance to the virus, so your husband not having cold sores (if he managed not to catch them from his mum as a child) doesn’t mean your daughters have the same risk profile. He could also be an asymptomatic carrier.
This isn’t a health anxiety problem, it’s a common sense problem. She’s shown she’s prepared to be careless with your daughters’ health and your health already because she feels she knows best. I would not be letting her kiss my kids either.
Ultimately you need to have confidence in your own boundaries, and explain to her what the consequences are of crossing them, and follow through. It sounds like you’ll be getting help for your anxiety, but if your MIL can’t understand and be supportive then your husband needs to step up and look out for you, at least until you’re through this difficult time. Her actions come from a place of love and delight but she needs to recognise that they’re not helping right now.